Preheat the oven to 375°F and lightly butter two mini cocottes.
Cook the mushrooms: Melt the butter in a large skillet over medium-high heat. Add the mushrooms, thyme, salt, and cracked black pepper and cook for 8–10 minutes, stirring occasionally, until the mushrooms have released their moisture and are browned and crisp around the edges.
Make the cream sauce: Reduce the heat to medium-low and add the heavy cream and balsamic vinegar. Stir to combine and cook for another 2–3 minutes, until the cream has reduced and thickened enough to coat the mushrooms.
Fill the cocottes: Divide the creamy mushroom mixture evenly between the two prepared mini cocottes, creating a small well in the center of each.
Add the eggs: Crack one egg into the center of each cocotte, then sprinkle the Gruyère around and over the egg.
Bake: Place the cocottes on a baking sheet and bake uncovered for 10–15 minutes, or until the egg whites are set and the yolks are cooked to your liking. Start checking around the 10-minute mark, as the eggs can cook quickly toward the end.
Finish: Remove from the oven and let sit for 1–2 minutes. Finish with chopped chives and freshly cracked black pepper.
Serve: Serve warm directly from the cocottes, with toasted sourdough alongside for dipping into the egg and creamy mushrooms.