3 1/2 cups All Purpose flour, plus more for kneading and rolling
4-6 extra-large eggs (use the freshest eggs you can find)
1/2 cup cooked red beets, puréed in a blender or food processor until smooth
1/4 cup semolina flour
3 garlic cloves, smashed
1/4 cup cured Calabrian chiles, split in half, seeds and stems removed
1/4 cup mint leaves, rustically torn
12 whole Calabrian chiles, reserved for garnish
3 cups cherry tomatoes, slow roasted in oven and peeled
2 tablespoons unsalted butter
1/4 cup grated Parmigiano Reggiano
1/4 cup grated pecorino Romano
Make the pasta dough: Add flour to the bowl of a stand mixer fitted with the dough hook. Start mixing on low speed. Add the eggs one at a time, mixing well between each addition. Then add the beet purée. Stop the mixer and use a rubber spatula to scrape down the sides of the bowl as needed, until the dough comes together. (If after adding the first 4 eggs the dough is still dry or not coming together, add the extra eggs one at a time, mixing well between each, until the dough comes together. If the dough feels too wet and slack, add more flour, 1 tablespoon at a time.)
Remove the dough from the mixer and turn it out onto a lightly floured work surface. Knead until the dough is soft and smooth, 4 to 6 minutes. Wrap the dough in plastic wrap and refrigerate for 1 hour.
Roll the pasta: If using a stand mixer to roll the pasta, attach the pasta roller attachment and set it to the thickest setting (number 1). If using a hand-crank pasta roller, set up the roller on the work surface and adjust to the thickest setting (number 1).
Remove the pasta dough from the refrigerator, unwrap, and place on a cutting board. Slice the pasta crosswise into 3 equal pieces. Set 2 pieces aside and cover with a kitchen towel so they don’t dry out. Dust the remaining piece of dough with flour.
Press the dough into a flat oval shape and run it through the pasta roller. Repeat until the dough forms a long rectangle and holds together nicely, about 4 times. Adjust the setting to number 3 and roll through again, taking care not to overstretch the dough. After passing the dough through 4 times, increase the setting to number 4. Run it through once more, then lay it on the cutting board and dust with some of the semolina. Cut the sheets into 10-inch rectangles and place on top of each other dusting with semolina between each layer.
Using the “spaghetti” shape cutter, carefully add each sheet through the cutter. Place each sheet in a small nest and refrigerate on a baking sheet, covered with a kitchen towel. Pasta can be made a day in advance and kept covered in the fridge.
Bring a large pot of water to boil and season with salt. While the water is heating, add 2 tablespoons olive oil to a sautée pan over medium heat. Add garlic and cook slowly until the garlic begins to dance and turn golden brown. Remove garlic, then add cut chiles and tomato to pan. Cook until oil gets a red hue. Add pasta to salted water and cook for 3 to 4 minutes. Remove pasta (be sure to save some of the pasta water) and add to sautée pan. Add butter, pasta water, if necessary, and mint. Remove from heat and stir in cheese until sauce coats the noodle. Divide evenly between four bowls. Garnish with whole chili and serve immediately.