Heat oven to 425°F. Toss squash with oil and 1/2 teaspoon salt on a sheet pan and roast for 30 to 35 minutes, turning once, until very soft and browned at a few edges.
Melt the 3 tablespoons butter in a small saucepan over medium heat. Add chopped sage and cook for 2 to 3 minutes, swirling, until the milk solids turn golden and smell nutty; immediately remove from heat.
Blend roasted squash with milk, broth, remaining salt, pepper and nutmeg until completely smooth, working in batches if necessary. Stir in all the sage butter and keep the sauce pourable.
Mix ricotta, egg and Parmesan in a bowl. Butter a 6-quart slow cooker and spread 3/4 cup squash sauce over the bottom; divide dry noodles into three groups and break to fit.
Layer one group of noodles, one-third of the ricotta mixture, one-third of the remaining squash sauce and one-third of the mozzarella. Repeat twice, spreading sauce over every noodle edge.
Cover and cook on low for 3 to 3 1/2 hours, checking at 2 1/2 hours, until a knife slides through the center noodles and the filling is 165°F. Switch off and rest uncovered for 20 minutes before cutting into eight portions.