Preheat the oven to 350°F and spray a 9x13-inch pan with cooking spray.
In a large skillet, melt 2 tablespoons of butter over medium heat. Sauté the onions, bell peppers, and ham until the onions are softened and translucent.
Combine the frozen hash browns, sautéed ham and vegetables, and condensed celery soup in a large bowl. Stir until the hash browns are coated.
Transfer the mixture to the prepared 9x13-inch pan. Sprinkle half of the cheese over the casserole and cover it with foil. Bake for approximately 45 minutes.
Remove the foil, add the remaining cheese, return to the oven uncovered, and cook for 15 more minutes. The cheese should begin to brown when the casserole is ready.