In a bowl, beat together eggs, water, salt, and pepper. Melt butter in a large skillet or omelet pan; pour in egg mixture.
Cook over low heat, pushing edges in toward center as mixture sets. Tilt and turn pan as necessary to allow uncooked portion to flow to the edge. When surface is set but still moist, slide omelet onto oven-proof plate or baking sheet.
In a bowl, combine hash, bell pepper, and chili powder; spoon evenly over top of omelet.
Bake in a preheated 400-degree oven for 10 minutes. Sprinkle with cheese. Bake 5 minutes longer. Fold omelet in half.