Place 1 lb. golden beets (about 3), peeled, cut into small cubes (about ⅓”), in a medium heavy pot, pour in water to cover, and season generously with kosher salt. Bring to a boil and cook 5 minutes (set a timer). Add 12 oz. orecchiette or other short pasta and cook, stirring occasionally, until beets are fork-tender and pasta is al dente (about 1 minute less than package directions; pasta will finish cooking in sauce). Drain, reserving 2 cups cooking liquid. Shake pasta and beets to remove excess water. Wipe out pot; reserve.
Step 2
Heat ¼ cup extra-virgin olive oil in reserved pot over high. (Make sure pot is completely dry to avoid spattering.) Cook 1 bunch sage, stems removed (about 30 leaves), stirring occasionally and trying to keep in a single layer, until crisp and oil is no longer sizzling, about 3 minutes. (It’s okay if some leaves break apart.) Using tongs or a slotted spoon, transfer to paper towels; season with salt.
Step 3
Cook 1 lb. hot Italian sausage, casings removed if needed, in same pot, breaking into small pieces and stirring with a wooden spoon, until browned and cooked through, about 5 minutes. Add 7 garlic cloves, finely grated, and ¼ tsp. crushed red pepper flakes; cook, stirring, just until garlic is fragrant, about 30 seconds.
Step 4
Add reserved beets, pasta, and ½ cup cooking liquid, then 5 Tbsp. unsalted butter, 4 oz. Parmesan, finely grated (about 2 cups), 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt, and 1 tsp. freshly ground black pepper. Toss vigorously, adding more cooking liquid a little at a time if needed, until cheese is melted and pasta is coated in a glossy sauce. Stir in 1 Tbsp. finely grated lemon zest, 1 Tbsp. fresh lemon juice, and remaining 2 Tbsp. extra-virgin olive oil. Season pasta with more salt and black pepper if needed.
Step 5
If pasta looks dry, stir in more cooking liquid and divide among bowls; top with fried sage leaves and more Parmesan.