Adjust the oven racks to their top positions and preheat the oven to 475 degrees. Toss the squash, tomatoes, garlic and 2 sprigs thyme with 2 tablespoons olive oil and 1/2 teaspoon salt in a large bowl. Spread the mixture on a baking sheet, turning the tomatoes skin-side up. Toss the mushrooms with the remaining 4 tablespoons olive oil and 1/2 teaspoon salt in the same bowl; spread on another baking sheet. Place the tomato mixture on the top rack and the mushrooms underneath and roast until the squash is tender, the tomatoes char and the mushroom edges are crisp, 40 minutes.
Meanwhile, combine the grits, broth, milk, remaining 2 sprigs thyme, 1 tablespoon butter, 1/2 teaspoon salt and the cayenne pepper in a large microwave-safe bowl. Microwave on high, uncovered, for 16 minutes, stirring halfway through. (The grits should be fairly thick; if they're not, microwave up to 2 more minutes.) Remove and discard the thyme.
Heat the remaining 4 tablespoons butter in a small skillet over low heat. Add the hazelnuts and cook, stirring frequently, until fragrant and slightly brown, about 4 minutes. Whisk in the brown sugar and vinegar and remove from the heat.
Divide the grits among 4 plates. Top each with a portion of the roasted vegetables and crumbled blue cheese. Drizzle with warm hazelnut butter sauce before serving.