Learn French: Nous mangeons à midi.

In French, midi (noon) and minuit (midnight) act as standalone times. You never add heures after them. You just say à midi (at noon) or Il est midi (It is noon).

Why is "à" used here instead of just "Nous mangeons midi"?

The preposition à means "at" when you are talking about time. Whenever you want to say that an action happens at a specific time on the clock, you must use à before the time.

Why is there an extra "e" in mangeons? Shouldn't the ending just be -ons?

Great catch! The verb is manger (to eat). Normally, the nous ending is -ons. However, if we wrote "mangons", the "g" would make a hard sound (like in "go"). To keep the soft "g" sound (like in "gel"), French keeps the "e" before the -ons, making it mangeons.

Can midi also just mean "the afternoon"?

While it can occasionally refer to the middle part of the day in casual speech, when telling time it strictly means 12:00 PM (noon). If you want to talk about the afternoon in general, you use the related word l'après-midi (literally "the after-noon").