Preheat the oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish.

Melt butter in a large skillet over medium heat. Cook and stir onion and garlic in hot butter until onion is soft and golden brown, about 10 minutes. Add squash and brown sugar. Continue to cook and stir until squash begins to brown on the edges but is still somewhat firm in the center, about 10 minutes. Scrape mixture into the prepared baking dish; pour in chicken broth. Wrap the dish tightly with aluminum foil.

Bake in the preheated oven until liquid has been absorbed and squash is tender, 45 to 50 minutes.

Meanwhile, toss together Gruyère cheese, Cheddar cheese, bread crumbs, thyme, and rosemary in a medium bowl until well combined.

Uncover the dish and sprinkle evenly with Gruyère mixture. Sprinkle Parmesan cheese on top. Return to the oven and bake, uncovered, until topping is lightly crunchy and browned, about 15 minutes.