Preheat the oven to 350 degrees F and butter a medium baking dish.
Quarter the squash and zucchini lengthwise and cut into 1/2-inch wedges. Barely cover with well-salted water in a saucepan with the carrots, bring to a boil, cover and cook over medium about 20 minutes, until very soft.
Meanwhile, cook the onion slowly in 3 tablespoons butter over medium-low heat until very soft, 6 to 8 minutes. Add the garlic for 1 minute and scrape into a mixing bowl. Melt the last tablespoon of butter and toast 1/2 cup of the crumbs until golden; reserve.
Drain the squash, mashing it a bit, and add to the bowl with the remaining 3/4 cup crumbs, the cheese, a good splash or two of hot sauce, and plenty of salt and pepper. Stir in the eggs.
Spoon into the dish, scatter the toasted crumbs on top and bake uncovered about 30 minutes, until golden and lightly firm in the center.