2½ cups all-purpose flour plus extra for kneading and rolling
½ cup semolina flour plus extra for dusting
4 large eggs
¼ cup plus 1 tablespoon extra-virgin olive oil, divided
7 garlic cloves, thinly sliced
6 ripe tomatoes, cored and finely chopped
¼ teaspoon red pepper flakes
½ cup roughly chopped pitted green olives (preferably Sicilian)
1 tablespoon roughly chopped capers
12 basil leaves, roughly chopped
1. Make the pasta: On a cutting board or large work surface, use your fingers to combine the flours and a pinch of kosher salt. Make a well in the center and crack the eggs straight into the well. Use a fork to beat the eggs and slowly work a little flour from the perimeter of the well into the eggs, adding more flour as you mix until the eggs are no longer runny. Add the remaining flour and begin to knead the dough until it comes together into a ball. Continue to knead the dough until the ball is silky and supple, 6 to 8 minutes (if the dough becomes very sticky, flour your hands and continue kneading). Wrap the dough in plastic and set aside to rest for 1 hour.
2. Attach a pasta roller to your stand mixer (or set up your hand-crank roller) and dust the roller element with flour. Set the roller to setting No. 1. Unwrap the ball and use your hands to press it into a rectangle. Slice off a 1-inch length from one end and flatten the strip with your hand. Turn the mixer on to medium speed and feed the strip through the roller. Repeat. Turn the pasta-width setting to No. 2, reflour the roller element and the dough, and repeat rolling the dough twice through the No. 2 setting. Continue, working your way through each setting until you have rolled it through setting No. 6 twice. (When the dough becomes very long and hard to handle, cut it in half so you have two shorter pieces.) Dust the strips with semolina flour and fold the bottom end of each strip up about 4 inches. Continue to fold until each strip is folded into a 4-inch rectangle.
3. Set the pasta roller attachment back to the No. 1 setting. Flour the roller element and start the rolling process over with each piece of dough, feeding it into the No. 1 setting of the roller element and rolling it through each setting twice until it has passed through setting No. 6 two times. Dust the dough strips with semolina and fold the dough up into a 4-inch rectangle. Place the dough on a semolina-dusted baking sheet and set aside. Slice off a 1-inch strip from the pasta dough square and repeat the rolling process until all of the dough is rolled into sheets (you should have about 8 sheets total).
4. Remove the pasta roller attachment and replace with the fettuccine attachment. Dust the fettuccine element with flour and run each piece of dough through. Return the fettuccine to the semolina-dusted baking sheet, twisting it into a loose nest. Repeat with the remaining dough strips and set the fettuccine aside.
5. Make the sauce: In a large, deep skillet set over medium heat, add ¼ cup of the olive oil and the garlic and heat together until the garlic is fragrant, stirring often, 2 to 3 minutes (if the garlic threatens to brown, reduce the heat to medium-low). Add the chopped tomatoes and increase the heat to high. Stir in the red pepper flakes and cook, stirring often, until the tomatoes are cooked down and when a wooden spoon is dragged through the sauce it leaves a trail that doesn't immediately fill in, about 5 minutes. Turn off the heat and stir in the olives, capers and basil.
6. Set a large pot of salted water over high heat and bring to a boil. Add the fettuccine and boil until al dente, 1 to 2 minutes. Drain through a colander and immediately turn out into the skillet with the sauce. Use tongs to toss together and finish with the remaining 1 tablespoon of olive oil. Divide among bowls and serve.