In a large pot, bring 2 cups of water and a pinch of salt to a boil. Reduce heat and slowly pour in cornmeal, stirring constantly. Cook, stirring occasionally, until the polenta pulls away from the sides of the pot and is thick and lump free.
Stir in the butter and season with salt and pepper, to taste. Stir in the Parmesan and vegetables.
Press the polenta into a small parchment paper-lined pan, to the desired thickness and chill in the refrigerator.
Preheat oven to 300 degrees F. Using a star cookie cutter, cut polenta into stars and arrange them on a greased cookie sheet.
Bake shapes for 10 minutes until lightly browned. Let cool, pack, and serve with the yogurt dip.