Heat a 12-inch nonstick sauté pan over medium-high heat. Add the butter and oil and swirl around the pan until the butter melts. Add the country ham and cook until it begins to brown, about 3 minutes.
Turn the heat to medium-low. Stir in the roasted red bell pepper and beaten eggs. After the eggs begin to set, stir slowly from edge to edge with a heat resistant rubber spatula until large curds form. This will take anywhere from 5 to 8 minutes. If they are setting too quickly, reduce the heat to low. While the eggs are still moist and slightly runny, fold in the grated cheese and season with pepper.