In a medium bowl combine all ingredients with a spatula until the mixture is smooth. Make ahead: mixture can be covered and chilled overnight.
For the mushroom filling:
Heat the butter in a large skillet over medium-high heat. Add the mushrooms, thyme, salt and pepper to taste, and cook until the liquid evaporates and the mushrooms are tender and brown, about 10 minutes. Remove to a food processor bowl and let cool. Pulse the mixture until the mushrooms are finely chopped.Add the ricotta, Parmesan, and egg yolk to the food processor bowl and pulse the mixture until just blended. Make ahead: mixture can be covered and chilled overnight.
For the meat filling:
Heat the oil in a medium skillet over medium-high heat. Add the garlic and meat and cook, breaking up the meat into small pieces, until browned and cooked through, 3 to 4 minutes. Drain off the fat and remove to a medium bowl. Let cool.Add the ricotta, Parmesan, egg, basil, salt and pepper to taste and stir until well combined. Make ahead: mixture can be covered and chilled overnight.
To make ravioli:
Take one rolled-out sheet of pasta dough and place on a well floured surface. Cover any other sheets of pasta dough with a clean towel.Place a straight line of teaspoonfuls of filling about 1-inch apart on the bottom half of the sheet of pasta. The amount of space between the teaspoonfuls will depend on how big you want your ravioli to be. Make sure not to use too much filling otherwise the ravioli may rip or burst while cooking. Gently fold over the top half of the pasta sheet. Press the pasta down around each mound of filling, making sure to squeeze out any air pockets. If the pasta won't form a seal, dip a pastry brush in water and brush the outside of the filling to create a glue for a better seal.Use a fluted pastry wheel or ravioli stamp to cut out ravioli from the pasta. You may need to trim away extra pasta from the edges. Place the ravioli in a single layer on a lightly floured baking sheet. Fill, seal, and cut out as many sheets of dough as you desire.You can cook the pasta in gently boiling salted water for about 2 minutes or until al dente or you can freeze the ravioli for later. Place the tray of ravioli in the freezer until frozen then remove to an airtight container. The frozen ravioli can go straight into gently boiling salted water for about 5 minutes, or until al dente.