So, here's an idea:
My daughter started to eat carrots a lot more once I gave in and passed over the carrot peeler to her independent self. Is it dangerous for an almost 4 year old to peel her own carrots? I think not so much. When she tells me she is hungry for a snack and wants some candy, cookies, chocolate chips, etc., I kid you not, I say "How about you peel your own carrot and eat it?" "Yeah! Yeah," replies she.
She's peeling her own carrots from here on out. Maybe I can buy her her own pink, sparkly carrot peeler . . .