Step 1Dump the flour onto a table or wooden cutting board and hollow out a well in the center with your hand. Pour the water into the well, then the oil. Using a fork as if you are scrambling eggs, gradually incorporate flour from the inner walls of the well into the liquid. Mix until you reach a pancake batter-like consistency, using a bench scraper to transfer the remaining flour, a little at a time, into the well. As dough comes together, form it into a loose, shaggy mass. Wash your hands, then move the dough to a clean surface, and scrape up and transfer any flour or dough bits left on the board
Step 2Begin kneading the dough by folding the far edge toward you and pushing the dough down and away with the heel of your hand. Rotate dough a quarter turn and repeat, continuing to turn, fold and push for 2 to 3 minutes. The dough will gradually come together into a semi-smooth mass. Clean the work surface of any dried dough flakes as you work and continue to knead dough gently until it is smooth and supple, and springs back when poked, about 10 minutes. Wrap the dough ball tightly in plastic wrap and refrigerate for at least 30 minutes, ideally for 24 hours.
Step 3Find a textured surface, such as a sushi mat, a tightly woven basket or even a place mat. What you are trying to achieve is a texture on the surface of the pasta in order to capture and grip the sauce.
Step 4Remove dough from refrigerator and slice off a 1-inch section, then slice that in half. Roll both pieces out to form snakes slightly skinnier than your pinkie. Cut tubes into segments about 2 inches long.
Step 5To begin shaping the strascinati, form your index, middle and ring finger into a hook. Place a section of the dough onto your textured surface. Maintaining even pressure, press and drag the piece of dough toward you and up, like an airplane taking off. The dough should spring up and form around the curve of your fingers.
Step 6Transfer the strascinati to a baking sheet covered in parchment paper and sprinkled with flour, making sure no pieces of pasta touch. Continue until all dough is used. Let the pasta dry slightly, about 30 minutes. If not using immediately, cover the baking sheet with plastic wrap and refrigerate up to 1 day.
Step 1Bring a large pot of lightly salted water to a boil.
Step 2In a large sauté pan over high heat, add the olive oil. When you see light smoke, add the broccoli to the top half of the pan and immediately season with ¼ teaspoon kosher salt, or to taste.
Step 3Stir in the crumbled sausage, and cook, stirring occasionally, until the fat is rendered and the sausage browns, about 1½ minutes. Stir in the sliced garlic and chile.
Step 4Add the crushed tomatoes and reduce the heat to a low simmer. Cook about two minutes to marry the flavors. Taste and adjust the seasoning as needed and stir in the parsley. Remove from heat.
Step 5Drop the strascinati into the boiling water. Boil until the pasta pieces float and remain on the surface about 30 seconds. (The pasta should cook 1½ to 2 minutes total). Remove the pasta with a slotted spoon and add to the sauce, along with a couple of tablespoons of the pasta water. Toss vigorously to coat the pasta and evenly combine.
Step 6Sparingly grate the cheese over the pasta. Serve immediately.