Heat 5 Tbsp. unsalted butter in a medium saucepan over medium, stirring often, until it foams, then just starts to brown, about 5 minutes.

Add ¼ cup (packed) sage leaves and cook, stirring constantly, until crisp, about 1 minute. Pour butter through a fine-mesh sieve set over a small bowl. Season sage in sieve with kosher salt. Set brown butter and sage aside separately.

Heat 2 Tbsp. extra-virgin olive oil in a medium Dutch oven or other heavy pot over medium-high. Cook 1 medium onion, finely chopped, stirring occasionally, until softened and golden brown around the edges, about 8 minutes. Add 6 garlic cloves, finely chopped, and cook, stirring, until fragrant, about 1 minute.

Add one 15-oz. can butternut squash purée, 2 Tbsp. double-concentrated tomato paste, ½ tsp. crushed red pepper flakes, and 1 tsp. Diamond Crystal or ½ tsp. kosher salt and cook, stirring often, until mixture is thickened and slightly darkened in color, about 5 minutes.

Pour 4 cups low-sodium vegetable broth and 1½ cups water into pot. Increase heat to high; bring to a boil. Add 12 oz. orzo; reduce heat and simmer, stirring often and scraping up any pasta stuck to bottom of pot, until tender and sauce is thickened, about 10-12 minutes (it will thicken more as it cools). Remove from heat and add 3 oz. Parmesan, finely grated (about 1½ cups), and reserved brown butter; stir vigorously to combine. Taste orzo; season with more salt if needed.

Ladle orzo into shallow bowls and top with reserved fried sage.