When my 20 year old was about 2 I got her to eat veggies by appealing
to her mischievous personality.
When I was preparing veggies (chopping carrots, mushrooms, broccoli,
etc) I would let her watch from a safe distance, standing on a chair at
the kitchen counter. I would chop the veggies and push them over to
her to place in a bowl or a pot.
Then I would put away the knife (real important) and then walk away
from the chopped veggies, leaving them well within grabbing distance of
my 2 year old. Maybe she was 3, but I know she was very small.
When I walked away from it to the other room for a minute, I would say,
"Now don't touch those veggies OK?" Since she was the type that loved
to touch when I said no, she would nod, I would leave and then she
would proceed to grab and stuff as much in her mouth as she could. I
would come back, notice her cheeks were bulging and then leave again,
giving her a second opportunity. She would get plenty of veggies this
way, and she loved helping me in the kitchen.
Another suggestion would be a low fat carrot cake without the frosting,
some pumpkin muffins, some zucchini bread?