Melt 2 tablespoons butter in a heavy saucepan over medium heat. Cook and stir onion and garlic in hot butter until onion is golden and soft, 5 to 10 minutes.
Stir sweet potatoes, 1 3/4 cups cream, acorn squash, thyme, kosher salt, and pepper into onion mixture. Bring to a boil, reduce heat to low, and simmer until potatoes and squash are tender, 20 to 25 minutes.
Preheat oven to 385 degrees F (195 degrees C). Butter an 8x13-inch baking dish with remaining butter.
Pour potato-squash mixture into prepared baking dish. Drizzle remaining cream over potatoes and squash; top with bread crumbs and Asiago cheese. Cover baking dish with aluminum foil.
Bake in the preheated oven until tender and hot throughout, 25 to 35 minutes. Remove aluminum foil and bake until top is golden brown, about 10 minutes more.