1. Bring broth to a boil in a small saucepan; reduce heat to low and cover to keep warm.
2. Heat oil in a small skillet over medium-high heat. Add sage leaves and cook until crisp and slightly shriveled, about 30 seconds. Using a slotted spoon, transfer to a plate lined with paper towels to drain, reserving oil.
3. Transfer 1 tablespoon sage oil to a large heavy saucepan and heat over medium-high heat. Add squash and cook, stirring occasionally, until slightly softened and browned, 5 to 6 minutes; transfer to a plate. Add remaining sage oil, reduce heat to medium, add onion and cook, stirring, until translucent, 6 to 7 minutes.
4. Add rice and cook, stirring constantly, until glossy, 2 to 3 minutes. Add 1/2 cup broth and simmer, stirring constantly, until absorbed. Continue adding broth 1/2 cup at a time, stirring constantly, until absorbed, about 18 minutes.
5. Add squash, sliced sage, salt and Parmigiano-Reggiano and cook, stirring, 1 minute. Top with crispy sage leaves.