Melt the butter in a large cast iron skillet over medium-high heat. Add diced kielbasa and cook for 3 minutes, until it begins to brown (look for lightly toasted edges).

Add the diced potatoes to the skillet. Cook without stirring for 5-7 minutes, so they crisp and turn golden brown on the bottom (watch for a sizzle and color change).

Stir in the diced onion and red bell pepper. Cook for another 5 minutes until the potatoes are tender and the edges look deep golden (stir enough to keep everything cooking evenly).

Make four wells in the hash and crack an egg into each well. Cover the skillet and cook until the eggs reach your desired doneness, about 4 minutes for runny yolks (the whites should be set while yolks stay liquid).

Season with salt and pepper and garnish with fresh chives. Serve directly from the skillet, aiming for a runny yolk that spreads over the crispy potatoes.