Preheat an oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish with butter.
Cook the sausage with the onion, green bell pepper, and garlic in a large skillet over medium heat, breaking the sausage into small pieces with the back of a wooden spoon as it cooks, until no longer pink, 7 to 10 minutes. Season with the oregano, chili powder, and cumin; allow to simmer another 5 minutes. Pour the tomato sauce into the mixture; simmer another 10 minutes.
While the mixture simmers, bring a large pot of lightly salted water to a rolling boil; cook the spaghetti at a boil until cooked through yet firm to the bite, about 12 minutes; drain and return pasta to the pot.
Whisk the eggs with the milk; add to the pot with the drained pasta and toss to coat. Transfer the pasta into the prepared baking dish; top with the sausage mixture. Scatter the Monterey Jack and Cheddar cheeses over the top of the pasta and sauce. Cover the dish with aluminum foil.
Bake in the preheated oven until the cheeses have melted and the casserole is bubbly, about 20 minutes.