Melt 3 Tbsp butter in a 12-inch cast iron skillet (or other heavy skillet) over medium-high heat.
Add bell pepper and onion and saute 3 minutes.
Toss potatoes in with the bell pepper mixture then spread into an even layer, season with salt and pepper to taste. Let cook, tossing only occasionally until starting to brown about 6 to 7 minutes.
Toss in ham and cook until potatoes are tender, about 3 minutes longer. Push potatoes to one side of the pan. Reduce heat to low.
Melt remaining 1/2 Tbsp butter in empty half of skillet then pour in eggs. Season with salt and pepper and cook and scramble until eggs are just set and no longer runny.
Toss mixtures together, sprinkle with cheese and toss just a few times.
