Melt 2 tablespoons of butter in a large skillet over medium-high heat. Add the mushrooms and cook until soft and caramelized, about 5 minutes.
Add the thyme, salt, and pepper to the mushrooms. Stir in the balsamic vinegar and cook until it coats the mushrooms, about 3-4 minutes. Remove from heat and let cool completely.
In a medium bowl, stir together the ricotta, fontina, and parmesan cheeses. Season with salt and pepper. Fold in the cooled mushrooms.
Lay out about 6 wonton wrappers. Spoon 1 tablespoon of filling into the center of each. Brush the edges with water and top with a second wrapper. Press to seal, removing air, then crimp edges with a fork. Keep remaining ravioli covered.
To make the butter sauce, brown 8 tablespoons of salted butter in a large skillet over medium heat, stirring until golden and toasted.
Add rosemary and garlic to the browned butter and cook for 30 seconds to 1 minute until fragrant. Pour in the wine and chicken broth and bring to a boil.
Season the sauce with salt and pepper. Cook for 5 minutes or until the sauce has reduced slightly.
Meanwhile, bring a large pot of salted water to a boil. Boil the ravioli in batches for 1-2 minutes or until they float. Drain and divide among bowls. Ladle sauce over ravioli and top with parmesan and parsley.