Heat oil and butter in a large nonstick frying pan with a tight-fitting lid over medium-high heat until oil is shimmering and butter is melted.
Add half of the steak, season generously with salt and freshly ground black pepper, and cook, stirring rarely, until steak is browned and crusty, about 3 minutes.
Remove from the pan with a slotted spoon and set aside. Repeat with remaining steak.
Add potatoes and 1 teaspoon kosher salt to the pan, season with freshly ground black pepper, and stir to coat.
Pat potatoes down into a single layer with a spatula, reduce the heat to medium, and cook, stirring occasionally, until potatoes are golden brown on the bottom, about 4 to 5 minutes.
Add onion and stir to combine.
Cook, stirring rarely, until potatoes are golden brown all over and just give way when pierced with a knife and onion is just softened, about 5 minutes.
Add reserved steak and any accumulated juices to potato-onion mixture and stir to combine.
Create 4 indentations, crack 1 egg into each and season with salt and freshly ground black pepper.
Cover the pan and cook until egg whites are set and yolks have reached desired doneness.