Preheat the oven to 400°F. Clean all mushrooms and discard hard stems. In a mixing bowl, combine all the mushrooms, olive oil, thyme, and garlic. Add salt and pepper to taste. Transfer to a sheet pan and roast for 15 minutes. Remove from heat, discard thyme and garlic, and cover to keep warm.
While mushrooms roast, prepare the grits. In a saucepan over medium heat, combine milk, 2 cups water, jalapeño, garlic, bay leaf, rosemary, and salt. Bring to a simmer, then remove from heat. Pour through a fine-meshed strainer into a heatproof bowl. Discard solids.
Return the liquid to the saucepan and bring it to a boil over high heat. Add grits, lower the heat to medium, and stir constantly until fully cooked and smooth, about 15 to 20 minutes. Add butter and mix well.
Add the goat cheese immediately before serving and mix well. Divide the grits among six plates, and spoon the roasted mushrooms over them.