Preheat oven to 375 degrees F (190 degrees C).
Thinly slice all of the squash into 1/4 to 1/8-inch-thick slices using a mandoline or a large sharp knife.
Heavy Cream (2 cups)
All-Purpose Flour (2 Tbsp)
Kosher Salt (1/2 Tbsp)
Ground Black Pepper (1/2 tsp)
in a medium saucepan. Cook over medium heat, stirring often, until hot and slightly thickened but boiling; about 5 minutes.
Layer about 1/2 of the
Butternut Squash (1)
Kabocha Squash (1)
Delicata Squash (1)
in a lightly greased (with butter) 12-inch round baking dish.
Top with 1/2 of the
Gruyère Cheese (1 1/2 cups)
followed by half of the cream sauce. Layer in the remaining squash (reserve a handful of squash for the top), top with the remaining half of the cheese and pour the remaining cream mixture evenly over the squash. Arrange the last pieces of squash over the top of the cream.
Cover with foil and bake until bubbly, about 1 hour.
Uncover, increase oven temperature to broil, and cook until the top begins to brown, about 4 minutes.
Remove from oven and let it stand at room temperature for 10 minutes.
Butter (1/4 cup)
in a small skillet over medium heat.
Working in small batches, fry a few
Sage Leaves (8)
until edges begin to curl and leaves just begin to brown. Transfer leaves to a plate lined with paper towels to drain and cool.
Arrange sage leaves on top of gratin before serving.