1. Make the pasta dough: pour the semolina flour onto a board and form a well in the center. Pour in the warm water and gradually whisk in flour from the well with a fork until it forms a thick paste.
2. Switch to a bench scraper: fold in the remaining flour, cut and compress the mixture into a rough dough, then knead by hand for 3-5 minutes until it becomes a smooth, springy ball.
3. Wrap tightly in plastic wrap and let the dough rest 30 minutes, then knead again for another 3-5 minutes until smooth. Reshape into a ball, wrap again, and refrigerate overnight for best texture.
4. The next day, let the dough come to room temperature, then cut off a quarter (one portion) at a time, keeping the rest wrapped. Roll into a rope about half an inch thick.
5. Fold the rope onto itself to form four equal strips, then cut into pieces about three finger-widths long, dusting lightly with semolina flour to prevent sticking.
6. Shape each piece into cavatelli by pressing three fingers over it and rolling back toward yourself with even pressure, forming a rolled, ridged shape. Toss finished pasta in semolina flour to prevent sticking.
7. Thinly slice the garlic. Cut the broccoli florets into small dice, discarding the stems. Squeeze the sausage out of its casing into bite-sized chunks.
8. Blend about half a jar of hot cherry peppers with enough of their packing liquid to form a smooth puree, for the hot sauce.
9. Heat a splash of oil in a saucier or wide pan over medium-high heat. Add the sausage in an even layer and let it brown undisturbed, then stir to break up large pieces and brown further. Remove sausage, keeping the oil in the pan.
10. Add the broccoli to the same pan with a little more oil and a pinch of salt; sear undisturbed on one side, then stir to brown evenly.
11. Add the garlic (with more oil if needed) and cook until it just begins to toast at the edges. Return the sausage to the pan.
12. Bring a pot of water to a boil and season with 2-3 tablespoons of salt. Add the cavatelli and cook for about 1.5-2 minutes, until they float.
13. Stir a few tablespoons of the cherry pepper puree into the sausage-broccoli pan, then deglaze with a splash of pasta water.
14. Transfer the cooked cavatelli directly into the sauce with a few more tablespoons of pasta water. Toss over heat, reducing the sauce until it thickens and coats the pasta.
15. Turn off the heat, loosen the sauce with a little more pasta water, then sprinkle in the grated parmigiano reggiano while tossing off the heat until melted, creamy, and glossy. Adjust consistency with more pasta water if needed.
16. Plate and serve immediately.
