Bring a large pot of water to a boil; add beans. Cook until tender, 6 to 7 minutes. Transfer to an ice-water bath; let cool 1 minute. Drain; squeeze beans from skins. Transfer 2 cups of the beans to a food processor. Add cheeses, mint, lemon juice, and 3/4 teaspoon salt. Process until smooth. Refrigerate filling at least 1 hour (up to 2 days).

Divide dough into 4 pieces. Dust 1 piece of dough with flour (cover remaining pieces with plastic wrap). Set the rollers of a pasta maker to the widest setting; roll dough through. Fold dough into thirds; pass through again, narrow side first. Repeat until smooth, 3 or 4 more times. Run dough through progressively narrower settings, using additional flour as needed, until very thin (at least 5 inches wide). Cut sheet crosswise into manageable pieces.

Dust 2 rimmed baking sheets with cornmeal; set aside. Place 1 piece of pasta sheet on a lightly floured work surface (keep unused pieces covered). Space heaping tablespoons of filling 3 inches apart on lower half of sheet.

Using a wet pastry brush, moisten pasta around each mound of filling. Fold top half of sheet over filling to meet edge; press around mounds to eliminate air and to seal.

Cut pasta into 2 1/2- to 3-inch squares using a pizza wheel. Brush away excess flour. Place ravioli on cornmeal-dusted baking sheets. Roll out remaining pasta dough, and repeat. (If serving that day, cover ravioli with plastic wrap, and refrigerate on baking sheets until ready to use. If making ahead, freeze on baking sheets until firm, about 1 hour, and then transfer to an airtight container; freeze until ready to use, up to 1 month.)

Bring a large pot of water to a boil; add salt. Add half the ravioli; gently stir once. Cook at a gentle boil until ravioli are just tender, 3 to 5 minutes. Transfer to a colander using a slotted spoon; drain. Cover to keep warm. Repeat with remaining ravioli.

Meanwhile, melt 4 tablespoons butter in a large skillet over medium heat. Reduce heat to low; cook butter until lightly browned, 3 to 4 minutes. Add half the ravioli to skillet; toss to coat. Using a slotted spoon, transfer ravioli to a platter; cover to keep warm. Repeat with remaining ravioli.

Cook remaining 4 tablespoons butter until lightly browned as above. Add reserved cup beans; cook over low heat until warm, about 20 seconds. Transfer ravioli to serving plates. Spoon beans and butter over ravioli. Sprinkle with cheese; season with salt and pepper.