MELT 3 tbsp butter in a large frying pan over medium. Add carrots, parsnips and turnips. Cook until golden brown, about 15 min. Stir in garlic and thyme. Add 1/2 cup broth. Boil, then reduce heat to medium-low and simmer, until no liquid remains, 2 to 3 min.
BOIL remaining 3 1/2 cups broth in a large pot. Reduce heat to medium-low, then whisk in cornmeal. Cook, whisking constantly, until thickened, 1 min. Remove from heat. Add remaining 2 tbsp butter and let stand until melted, 1 min. Stir in milk and parmesan. Transfer to a platter and top with vegetable mixture.