For the pasta dough:
Pour the flour into a mound on a clean flat work surface. Make a well in the center of the flour.
Pour the water into the well. Slowly start dragging the edges of the flour into the center of the well with a fork to combine the flour with the water until the dough starts to come together. Form the mixture into a ball with your hands and knead until smooth, about 10 minutes. Cover the dough with plastic wrap or a towel and let it rest for at least an hour.
For the busiate:
Generously dust a large sheet pan with flour or cornmeal. Remove the dough from the plastic wrap and cut into four pieces of roughly equal size. Work with one piece at a time and keep the rest covered with plastic wrap.
Roll the dough with a rolling pin until it is about 1/2-inch thick. Then cut the dough into strips lengthwise that are about 1/4-inch wide. Roll one of the strips with two hands and with slight pressure until it forms a rope that is about 1/4-inch thick. Cut the rope into pieces that are about 5-inches long. Take one of the 5-inch pieces and lay it on a work surface vertically.
Take a bamboo skewer and angle it to about the 2 o’clock position. Press the top tip of the pasta piece around the skewer. Roll the skewer toward you so that the piece of pasta wraps around the skewer to form a helix shape. Lightly press the ends of the skewer onto the surface and roll gently to seal. Do not press too hard on the pasta piece or it will stick to the skewer. Slowly and lightly twist the pasta piece off the skewer. Transfer the shaped busiate to the sheet pan. Repeat the forming and rolling with the remaining dough. Let pasta dry for at least 2 hours on the baking sheet. While the pasta is drying, make the sauce.
For the sauce:
Preheat oven to 300°F.
Place the tomatoes, garlic, shallots, olive oil, spicy oil, and salt in a Dutch Oven or large pot and stir to combine. Place pot in preheated oven and roast for 1-1 1/2 hours until the tomatoes are cooked down and the garlic is golden brown. Remove from the oven and let mixture cool slightly.
Carefully pour the tomato garlic mixture into a blender or food processor and blend until smooth. Pour the sauce back into the Dutch Oven or pot.
When pasta is dry, fill a stockpot 2/3 full of water. Salt the water heavily and bring the water barely to a boil. Add the dried busiate to the boiling water, and stir to be sure pasta does not stick to the bottom of the pot. Cook for about 5 minutes until al dente, or desired texture. Reserve about 1 cup of the pasta water and then drain the busiate using a colander.
Add 1/2 cup of pasta water to the tomato garlic sauce in the pot, and set pot over low heat. Whisk the water into the sauce, adding more water if needed to create a smooth, creamy sauce. Add the busiate to the sauce and stir until pasta is completely coated. Add the arugula and basil to the sauce. Turn off the heat and stir until combined. Season to taste with salt and pepper. Serve pasta garnished with the pecorino cheese.
