Preheat oven to 350 degrees F (175 degrees C). Lightly oil a baking dish large enough to hold the 4 squash halves in one layer.
Crumble sausage into a bowl. Add salt, cumin, black pepper, cinnamon, diced pepper, buttermilk or yogurt, chopped mint, and egg. Mix well, breaking mixture down to a paste-like consistency, 3 or 4 minutes. Mix in cooked rice.
Divide mixture among the squash halves. Sprinkle stuffing with a pinch or 2 of breadcrumbs and some grated cheese. Drizzle lightly with olive oil.
Cover dish loosely with aluminum foil, more tented than wrapped; foil shouldn't touch the tops of the stuffed squash.
Bake in preheated oven for 1 hour. Remove foil and bake until top is golden brown and squash is soft and tender, 30 to 45 minutes.