Heat oven to 350ºF.
Heat the oil in a saute pan. Add the onion, carrots, and garlic; cook 6-8 minutes, or until soft. Add the mushrooms and cook another 2-3 minutes.
Peel the eggplant, then use the coarse side of a box rater to shred it. Toss the shredded eggplant with 1 teaspoon kosher salt. Use your hands to squeeze out as much moisture as you can.
In a mixing bowl, combine the eggplant, cooked vegetables, ½ cup bread crumbs, egg, and red pepper flakes. The mixture will be moist, but should hold together – add more breadcrumbs if needed.
Roll the eggplant into 24 balls, then roll each ball in the remaining breadcrumbs. Arrange in a single layer in a baking pan.
Cover with marinara sauce and top with cheese. Cover the pan with foil and bake for 20 minutes. Uncover and bake an additional 10 minutes. Garnish with parsley.