01 -
Melt the butter in a large skillet over medium heat. Add the diced kielbasa and cook until the edges brown and the fat starts to render, about 4-5 minutes.
02 -
Add the onion to the same pan and lower the heat. Cook slowly, stirring occasionally, until soft and golden, 6-8 minutes.
03 -
Crack the eggs into a bowl with a pinch of salt and pepper and whisk with the milk, if using, just until combined. Don't overbeat.
04 -
Pour the eggs into the pan over low heat. Stir gently and often, pulling curds from the edges toward the center, until soft curds form, 3-4 minutes.
05 -
Pull the pan off the heat while the eggs still look a little glossy. Scatter chives on top and serve immediately with rye bread.