Heat the skillet over medium heat and add the butter and olive oil.
Cook the butter, stirring occasionally, until it foams and turns golden-brown with a nutty aroma, about 4–6 minutes; watch closely to avoid burning.
Add the chopped onion and cook until softened, about 3–4 minutes, then stir in the minced garlic and cook 30–60 seconds until fragrant.
Pour in 1/2 cup broth and bring to a simmer, scraping up any browned bits from the pan.
Add the tortellini to the skillet, stirring to coat; add additional broth as needed so pasta is partially submerged, cover, and simmer until tortellini are heated through and tender, about 6–8 minutes depending on package instructions.
Uncover, stir in the fresh spinach until wilted, about 1–2 minutes; season with salt, pepper, and red pepper flakes if using.
Remove from heat and stir in grated Parmesan; taste and adjust seasoning, then serve immediately.