Preheat oven to 400F; line a baking sheet with foil.
Cut top and bottom off a 2 1/2 lb. butternut squash and halve it lengthwise; remove seeds. Put halves cut side down on pan; bake for 1 hour, until squash is fork-tender.
Pulse 1 1/2 cups squash flesh, 3 eggs, 1 cup flour, 1/3 cup grated Parmesan, 1 tsp. salt and 1/4 tsp. pepper in a food processor until smooth.
Bring a large pot of salted water to a boil. Prepare a large bowl of ice water. Using a spoon, scoop 1 Tbsp. squash mixture. With another spoon, form an oval; drop it into boiling water. Make 10 gnocchi. When they float, cook for 5 minutes and then place in ice water. Repeat with remaining batter.
Let cool and pat dry.
Melt 4 Tbsp. unsalted butter over medium heat.
Add 1 chopped shallot; cook, stirring occasionally, until soft, about 5 minutes.
Add gnocchi and 2/3 cup vegetable broth; cook, turning until heated.
