Heat olive oil in a large stainless steel or cast-iron skillet over high heat until very lightly smoking. Immediately add squash, season with salt and pepper, and cook, stirring and tossing occasionally, until well-browned and squash is tender, about 5 minutes. Add butter and shallots and continue cooking, stirring frequently, until butter is lightly browned and smells nutty, about 1 minute longer. Add sage and stir to combine (sage should crackle and let off a great aroma). Remove from heat and stir in lemon juice. Set aside.
In a medium saucepan, combine pasta with enough room temperature or hot water to cover by about 2 inches. Season with salt. Set over high heat and bring to a boil while stirring frequently. Cook, stirring frequently, until pasta is just shy of al dente, about 2 minutes less than the package directions. Drain pasta, reserving a couple cups of the starchy cooking liquid.
Add pasta to skillet with squash along with a splash of pasta water. Bring to a simmer over high heat and cook until the pasta is perfectly al dente, stirring and tossing constantly and adding a splash of water as needed to keep the sauce loose and shiny. Off heat, stir in Parmigiano-Reggiano. Adjust seasoning with salt and pepper and texture with more pasta water as needed. Serve immediately, topped with more cheese at the table.