1. Make the Pesto: Place garlic on a cutting board and sprinkle with a pinch of salt. Crush using the flat side of a large knife. Continue crushing using both sides of the knife in a back-and-forth fashion until a paste is formed. Transfer garlic paste to the bowl of a food processor along with basil and both cheeses; process until smooth. Add pine nuts and pulse until almost smooth. Add olive oil and butter; let stand for about 3 minutes and season with salt and pepper. Pesto may be stored in an airtight container, refrigerated, for up to 3 days, or frozen for up to 1 month.

2. Make the Triofe: Place semolina on a work surface and make a well in the center. Add 6 tablespoons water to well. Slowly incorporate the semolina with the water until well combined (if mixture seems to dry, add water, one teaspoon at a time, as necessary); knead to form a dough. Cover and let stand for 15 minutes.

3. Working on a large clean wooden board, break off a walnut-sized piece of dough and roll to make a long, 1/8-inch-thick strand; cut each strand into 1 1/2-inch-long pieces.

4. Working with one piece of dough at a time, use the flat side of a knife at a 45-degree angle to both the board and the dough, and draw it across and along the dough to create a deep, spiraling groove along the length. Grooves should extend within the dough, and inner surface will be rough and outside smooth. The dough has now been rolled and stretched into a rectangle in the process, the angle of the knife causing the dough to roll up diagonally, creating the torpedo shape. Repeat process with remaining pieces. Repeat process until all the dough is used; set aside and let dry slightly, about 20 minutes.

5. Alternatively, roll one piece of pasta between your hands to start the torpedo shape. One by one, roll the lengths between the ball of your palm and the board with considerable pressure, sliding your hand along the length as you go. The friction should cause the strip to flatten and twist into an irregular helix. Add a little more pressure at the start and finish to taper the ends and create a torpedo shape. You need to have friction between the dough and the board; if dough skids, dampen board slightly. Repeat process with remaining pieces. Repeat process until all the dough is used; set aside and let dry slightly, about 20 minutes.

6. Bring a large pot of water to a boil over medium-high heat; generously salt and return to a boil. Add pasta, potatoes, and green beans; cook until pasta, potatoes, and green beans are tender, about 5 minutes. Drain and serve with a dollop of pesto.