Make the ravioli: Dust a baking sheet with semolina or cornmeal. Work with each sheet of dough just after it's been rolled. Cut sheet in half crosswise, and trim each half to a 3-inch-wide strip. Space tablespoons of filling 3 1/2 inches apart along center of 1 strip. Top with remaining strip, and gently press around filling to seal, working from center out. Using a fluted pastry cutter, cut into 3-inch squares. Transfer ravioli to sheet, and cover with a kitchen towel. Repeat with remaining pasta dough and filling. (To store, lightly dust ravioli with semolina, and refrigerate between layers of parchment in an airtight container for up to 4 hours. Alternatively, freeze ravioli in a single layer on a baking sheet, about 15 minutes, and then pack as described above; freeze for up to 1 month. Do not thaw before cooking.)

Bring a large pot of water to a rolling boil. Add salt and half of the ravioli. Cook, stirring occasionally to separate, until edges are just tender, about 4 minutes. Using a slotted spoon, transfer to a colander to drain. Drizzle with oil, and toss gently to coat. Transfer to a bowl, and loosely cover to keep warm. Repeat with remaining ravioli.

Meanwhile, melt butter in a small saucepan over medium heat. Cook until butter is dark golden brown and has a nutty aroma, about 4 minutes.