Preheat oven to 375º. In medium frying pan, melt half the butter. Add shallots, 1/4 teaspoon of sea salt, and half a tablespoon of fresh thyme leaves. Sauté over medium heat about 10 minutes, stirring frequently until shallots are translucent and just beginning to darken. Add remaining butter to the pan, then add mushrooms and sauté another 3-5 minutes, until mushrooms are tender but not too soft. Remove from heat.
In a large bowl, whisk eggs, milk, black pepper, nutmeg, half a tablespoon of fresh thyme leaves, and 1/4 teaspoon sea salt. Whisk until fully combined.
Lightly butter a 9” round pie pan (or 8” square baking dish, or rectangular dish of a similar sizand pour in egg mixture. Sprinkle in the sautéed shallots and mushrooms, and crumble in the goat’s cheese—distribute all evenly throughout.
Bake for 25-30 minutes, or until golden brown around the edges, poufy, and no longer jiggly in the middle. Let cool 5-10 minutes, and then enjoy with a light drizzle of balsamic vinegar or balsamic reduction on top!