Place pumpkin, along with 3/4 cup water, in a large pan over medium heat. Cover and steam until tender, 20 minutes. Remove to a medium bowl and mash until smooth.
Heat 1 1/2 tablespoons olive oil in a small skillet and sauté garlic until golden. Transfer garlic to a mortar and pestle, then crush to a paste. Stir cheeses, chopped sage, salt, pepper, and garlic paste into mashed pumpkin. Set aside.
Bring a large pot of water to boil. Cook lasagna sheets until tender, about 2 minutes. Transfer to a plate and drizzle with olive oil to prevent pasta from sticking together. Reserve 1/4 cup pasta water.
Liberally brush a medium baking dish with oil. Place a lasagna sheet on a clean work surface. Add 4 tablespoons pumpkin mixture to centre of lasagna, roll into a cannelloni tube, then transfer to prepared baking dish. Repeat with remaining sheets.
Pour reserved pasta water over lasagna and cover tightly with foil. Bake until heated through and pasta is tender, 20 to 25 minutes. Cook butter and sage leaves in small skillet over medium-high heat until golden-brown. Drizzle over pumpkin cannelloni and serve immediately.