Since it's important for health to consume some of the green stuff, it's helpful to find out how you dislike it least. For instance, cooked spinach is just as revolting as raw, but you only have to swallow about a quarter of the amount because of shrinkage. Broccoli and Brussels Sprouts are better if you roast them until they mostly taste like brown. And shrinkage happens there, too.

Zucchini is a green vegetable that's less offensive imho, and can hide in a spaghetti sauce without ruining it.

Don't go with any recipe that "celebrates the delicious flavor" of anything you don't already like.

You can also be your own mom and puree veggies into the gravy/sauce of your meat.

Anyway, good on you for eating the veggies first. It'll also start filling you up, so the steak will feel more satisfying.