Pureed veggies can make a great pasta sauce. I used to give DD mini macaronis with pureed squash/sweet potato/carrot and parm cheese, and she completely loved it. It looks just like mac and cheese, but better. I haven't made her that in a while, actually . . .
There are also lots of recipes around for broccoli or other veggie nuggets - maybe try that? You can also buy broccoli nuggets, and that might be better before you spend the time making a whole batch only to find out your LO doesn't like them either. Or use a veggie puree as a dip for regular chicken nuggets? Does he like veggie burgers or veggie nuggets?
I assume you have already tried everything, like cooking them lots of different ways - raw, steamed, grilled, roasted, etc? All different kinds of seasonings? Mixing different things together, like carrots and turnips?
I do a smoothie in the morning with spinach in it. You can not taste the spinach at all. I have been doing that just in case he doesn't eat any other veggies through out the day but I have found if he sees me eating veggies he is more likely to try it. I usually take a bite and then offer him a bite of the same one I just had.
One cup spinach
2 cubes of yorgurt (I freeze my yogurt in ice cube trays)
One cup of frozen fruit
1/2 cup juice or milk
Blend the spinach in the milk or juice before adding anything else and less likely to get spinach chucks.
I add chopped spinach and grated carrots to my meatballs and meatloaf. LO loves baked sweet and white potato fries - I just take potato and slice into sticks, toss in olive oil, salt and pepper and bake for ~15-20mins until tender at 400.
At that age LO loved peas - I think the challenge of picking them up part of the joy. I would try adding finger food veggies, so broccoli spears, etc. LO may enjoy eating them if he can feed himself. FWIW, my kid has never liked mashed potatoes. He'll eat baked sweet potato but not mashed. I don't get it lol.
And lastly, remember a serving size for an 11m old is like a tbsp. So, a bite or two. They don't need to eat a big pile of green beans, or a bowl of mashed potatoes. Keep offering different things and let LO self-feed and you may be surprised how much makes it in.