Bring large pot of water to boil and add salt until it tastes like the sea. Add the pasta and cook to just shy of al dente (according to package directions).
Meanwhile, heat a small glug of olive oil in a large skillet over medium heat. Add sausage patties and cook until nicely browned on one side, about 4 minutes.
Add broccoli coins and the sliced garlic, including the oil, to the skillet. Flip the sausage patties and keep cooking until the sausage is just about fully cooked (it's OK if it's a touch pink in the center because it will continue to cook a bit), another 4 minutes or so. Break up sausage with a spoon into bite-size chunks.
Add chile flakes and cook for 30 seconds or so. With a ladle or measuring cup, scoop out about 1/4 cup of the pasta's cooking water, add it to the pan to stop the cooking of everything, and slide the pan from the heat.
About 3 minutes before the pasta should be al dente, add the broccoli florets and cook all together until pasta is ready. Scoop out another cup of pasta cooking water, drain pasta and broccoli, and add to the skillet.
Return skillet to heat. Add 1/4 cup or so of the pasta water, the Whipped Ricotta, and half the Parmigiano. Season generously with salt and black pepper. Shake the pan to combine the ingredients, put back over medium heat, and cook for a couple of minutes to warm everything through and make a nice saucy consistency. Serve with more Parmigiano and top with breadcrumbs, if desired.