What can I call this Swiss Chard side dish exactly in a menu?

In the U.S., we would call usually these greens and not herbs. So why not use "sauteed field greens" or "sauteed wild greens"?

You should use a word like sautéed to indicate that they are cooked, because otherwise people will expect a salad.

If the menu actually provides descriptions of dishes, then I would just use the Italian name and describe it in English. This is very normal at least in the US for dishes from international cuisine, and even many ‘local’ dishes here in the US have associated descriptions on their menus unless they are something truly trivial and universally recognizable (and even then, they may still have such descriptions, a ‘cheeseburger’ sounds simple, but there’s enough variety that most places will explicitly list the toppings and the size of the patty)

Barring that possibility, I would go with the admittedly somewhat long ‘Field greens sautéed with garlic and chilli flakes.’. My reasoning here is as follows:

• The generic term for leafy green vegetables foraged from a field is ‘field greens’ (you can add ‘foraged’ to the beginning and charge an absurd premium if they’re actually foraged, but this is not required). It’s also important at least with some dialects to not use the term ‘herbs’ as that implies stuff like dill, basil, oregano, chives, or oregano (that is, ‘herbs’ in the culinary/medicinal sense, not in the botanical sense).
• ‘Cooked’ is very generic and potentially misleading, while ‘sautéed’ describes exactly how they are cooked (this matters, steamed or baked vegetables are still ‘cooked’, but they are very different from sautéed vegeetables).
• ‘Sautéed’ implies the use of some type of fat, and most people will not care what exactly that fat is (if they do, they will want more specificity than ‘oil’ because they will probably be looking to avoid specific fats). Also, given that it’s Italian, it’s kind of implicit (at least in most parts of the US) that it’s oil (as opposed to lard or butter).
• Garlic is kind of important as a flavor component here (I’m assuming this is some variation of erbe di campo, which I’ve made myself before a couple of times, and believe me, the presence or absence of garlic has a big impact on the flavor) so it’s worth mentioning. Calling it out also helps reinforce that this is not just some kale or chard tossed in a skillet for a bit and then served as-is.
• It’s important to call out spicy (as in piccante, not speziato) flavors, as many people are sensitive to them, hence mentioning the chili. Additionally, calling out ‘chili flakes’ instead of just ‘chili’ is important in some dialects, as just ‘chili’ may imply ‘chili powder’, which is not just ground up dried chili pepper in some parts of the world (in the US, ‘chili powder’ is the name of the spice mix used as a base for chili con carne, and almost always includes at least cumin, garlic, and onion in addition to the ground chili peppers).

Why not "herbe di campo?" If it has to be in English, "sauteed wild herbs" can also work...or "sauteed herbs." Much of this depends on the context of how the rest of the menu is written.

Are you the chef? If not, have you asked the chef. I would call this dish "Seasoned Field Greens."