In the same skillet over medium heat, melt 2 Tbsp. butter. Add the diced onion; season with ½ tsp. salt. Cook, stirring occasionally, until the onion is translucent, about 7 to 9 minutes.
Add garlic and sage and cook, stirring, until fragrant, about 1 minute more.
Add orzo and the remaining 2 Tbsp. butter and cook, stirring occasionally, until toasty and golden brown, 2 to 3 minutes.
Add 2 ⅔ c. chicken OR vegetable stock, then bring to a simmer over medium-high heat. Reduce heat to medium-low; cover and simmer until orzo is tender and liquid is absorbed, 7 to 8 minutes.
Meanwhile, reserve 1 c. of the cooked squash cubes. Transfer the remaining squash cubes to a blender or food processor. Add 3 Tbsp. of the chicken or vegetable stock and blend until smooth and creamy.
Add pureed squash to the orzo mixture and stir until creamy and combined.
Fold in Parmesan, ¼ tsp. pepper, and reserved squash and cook, stirring, until cheese is melted and squash is heated through, about 3 minutes; season with salt, if needed.
Divide the orzo mixture among bowls. Top with more Parmesan and black pepper.
OPTIONAL: Serve with butter-fried sage leaves on top.