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made in house

Another answer mentions that the French and Italian equivalents to in house are often used in English menu-ese - and the straight English version is itself used too.

In house isn't food-specific, as the following definition from Oxford Living Dictionaries shows:

in-house adjective

attributive Done or existing within an organization.

‘in-house publications’

‘Each employee receives ten days of training a year, as well as an in-house training and development programme called Jigsaw.’

‘Yet there are at least a dozen reasons why organizations still prefer in-house application development and deployment.’

...

The house in in-house is, I presume, the same house as we find in phrases such as house red, house speciality, full house, on the house and one sense of house rules; the relevant definition and sub-definitions from Oxford Living Dictionaries follow:

house noun

...

2 A building in which people meet for a particular activity.

‘a house of prayer’

2.1 A business or institution.

‘he had purchased a publishing house’

2.2 A restaurant or inn.

as modifier ‘I ordered a bottle of their house wine’

‘Food arrives at our table - not food we have asked for, but a small
appetiser with the compliments of the house.’

‘We'll dine at the fanciest and snootiest drive-thru restaurants and
waffle houses.’

...

A few food-relevant examples found in the wild follow:

From the August 2018 menu of The Grace, Bristol:

Everything made in house

There is adjectival use of the phrase in the following headline:

Zuuk Mediterranean Brings Bold Flavorful Bowls Paired with Addicting Made-in-House Dips to Brickell

Foodable Network, Kerri Adams, September 2, 2017

...

Enter Zuuk Mediterranean.

The first store in Brickell opened in April and serves Mediterranean-inspired items. This style of food appeals to today’s diners (millennials, in particular) because it offers an array of flavorful healthy choices.

...

Zuuk has all the elements that the market currently demands including variety of flavors, affordability, and fresh ingredients sourced daily and cooked in-house. All of which is in a fun and friendly ambiance.

The relevant parts of a New York Times article follow:

Made in House? Prove It

New York Times, Elaine Sciolino, July 22, 2014

PARIS — The black-and-white symbol looks like a saucepan with a roof
for a lid. And if it sits next to an entry on a restaurant menu, it
signals that the dish is “fait maison” — house-made.

Or does it?

A new consumer protection law meant to inform diners whether their
meals are freshly prepared in the kitchen or fabricated somewhere
off-site is comprehensive, precise, well intentioned — and, to hear
the complaints about it, half-baked...

This article also uses the phrase house-made, which, as another answer points out, is a valid North American alternative to made in house.

A bit of a tangent regarding homemade

Also in The New York Times article, homemade is used, as mentioned in another answer, as equivalent to made in house.

This use of home made not meaning made at home but made on the premises, strikes me as American rather than British, and there is some evidence to back that up.

Oxford Living Dictionaries defines home-made as:

adjective Made at home, rather than in a shop or factory.

Whereas the quintessentially American Merriam-Webster gives the following definition:

adjective

1 made in the home, on the premises, or by one's own efforts

2 of domestic manufacture

(I've added italics to highlight the relevant part of the Merriam-Webster definition.)

There does, indeed, appear to be a transatlantic difference here, although I imagine any British English speaker would be familiar enough with the American meaning not to even pass remark on it.

The standard English answer would be "home made" (dictionary.com, which uses the example of "the restaurant's pastries"). The space is optional, so you could sell "homemade pizza".

"Maison" (French for house) is sometimes seen even in English, and an Italian restaurant may well use "della casa" ("of the house").

"From the oven" doesn't really work, as it doesn't say who made it before putting it in the oven. If the oven is a traditional wood-fired oven, that would be worth boasting about, and could be used to imply homemade as well as good quality.

A commonly used idiomatic expression is (made) from scratch.

Pizza made from scratch!

TFD(idioms):

from scratch

Entirely without the aid of something that is already prepared or in
existence. Refers to making something, usually food, from the raw or
base ingredients or components, rather than those that have been
preassembled or already partially completed.

She doesn't have time to make cupcakes from scratch, so I'm sure
they're from a box.

If you want some real from scratch cooking, try
Jesse's Café—it's as close to homemade as it gets.

Farlex Dictionary of Idioms. © 2015 Farlex, Inc, all rights reserved.

If you're going for brevity, how about "pizza made here" or "fresh pizza"?

"Made on premises" is common short-hand.

I'd use "house-made" in American English.

Oxford Living Dictionaries:

house-made

North American - (of food or drink served in a restaurant, cafe, etc.) made on the premises.

Anecdotally, I've seen this frequently used in restaurants in the US to refer to food made by the restaurant. I'd expect it to be understood by a general audience to mean "made in-house", and it's a bit more versatile for use in a blurb.