Preheat the oven to 450˚. Make the mushroom broth: Combine 7 cups water, the fresh mushroom stems, dried mushrooms, bay leaf and thyme sprigs in a medium saucepan. Bring to a boil, then reduce the heat to a steady simmer and cook until the liquid is reduced by almost half, about 20 minutes. Strain through a fine-mesh sieve into a large liquid measuring cup. (You’ll need 4 cups broth; add more water if needed.) Wipe out the pan and return the broth to the pan.
Meanwhile, combine the squash, 2 tablespoons olive oil, 2 grated garlic cloves, the thyme leaves, 1 teaspoon salt and a few grinds of pepper in a large bowl; toss. Spread on a baking sheet and bake until the squash is tender and lightly browned, about 30 minutes.
Return the mushroom broth to a simmer. Gradually add the grits, whisking constantly. Season with 1 teaspoon salt. Reduce the heat to medium low and cook, stirring occasionally with a wooden spoon, until tender, about 20 minutes. Stir in the heavy cream and bring to a simmer. Stir in the butter, fontina and parmesan. Season with salt and stir in the parsley. Remove from the heat; cover to keep warm. If the grits seem too thick, thin with water, 1 to 2 tablespoons at a time.
Meanwhile, heat the remaining 3 tablespoons olive oil in a large skillet over high heat. Add the sliced mushroom caps and season with salt and pepper. Cook, undisturbed, until lightly browned around the edges, about 3 minutes. Stir and continue cooking until browned and tender, 3 to 5 more minutes. Push the mushrooms to the edge of the skillet. Add the shallot and remaining grated garlic clove to the center and cook until softened, about 1 minute. Stir everything together and season with salt and pepper.
Divide the grits among 4 shallow bowls. Top with the mushrooms and squash. Sprinkle with chives.