1. Dampen a doubled, large piece of cheesecloth with cold water and use it to line a fine-mesh strainer. Set over a medium bowl, add the ricotta and refrigerate for 1 hour. Discard the excess liquid and reserve the ricotta.

2. In a medium skillet, add the olive oil, garlic and red pepper flakes. Place the skillet over medium heat and cook until the garlic is lightly browned, 4 to 5 minutes. Add the mushrooms and white wine and bring to a simmer. Cook until the wine has evaporated and the mushrooms are tender, about 8 minutes. Season with salt and pepper and transfer to a medium bowl.

3. Make the balsamic brown butter: In a medium saucepan set over low heat, add the butter. Melt the butter, stirring frequently, until it is fragrant and turns a deep brown color, 10 to 12 minutes. Remove the pan from the heat and set aside to cool for 5 minutes, then stir in the balsamic vinegar. Season with salt and pepper.

4. To finish: Place the oven rack in the top of the oven and preheat the broiler to high. Divide the strained ricotta between four shallow ramekins, slightly mounding the ricotta in the middle of each dish. Place the ramekins on a rimmed baking sheet and place under the broiler. Cook, rotating the baking sheet halfway through cooking, until the ricotta is browned on top and heated all the way through, 5 to 7 minutes. Sprinkle with salt and pepper and top with the mushrooms. Drizzle with a little balsamic brown butter (store the rest in an airtight container for up to 1 week). Serve the baked ricotta warm with bread.