Set an oven rack about 6 inches below the broiler and heat the broiler. Put a 10-inch oven-safe skillet on the stove so it is ready to use.

Pour the olive oil into the skillet and warm it over medium heat for about 1 minute.

Add the smoked sausage to the skillet in one layer. Cook for 4 to 5 minutes, turning once, until the edges brown lightly.

Transfer the sausage from the skillet to a plate and keep the skillet on the stove over medium heat.

Add the onion to the skillet with 1/4 teaspoon of the salt. Cook for 3 minutes, stirring often, until the onion softens.

Add the potatoes to the skillet and stir them with the onion. Pour in 1/3 cup water, cover the skillet, and cook for 6 to 8 minutes, until the potatoes are starting to turn tender when pierced with the tip of a knife.

Uncover the skillet and cook the potatoes for 2 to 3 minutes more, stirring once or twice, until the water has cooked away and the potato edges pick up a little color.

Add the cabbage to the skillet with another 1/2 teaspoon of the salt and all of the black pepper. Cook for 4 to 5 minutes, stirring often, until the cabbage wilts and the pan looks mostly dry again.

Return the browned sausage to the skillet and stir it into the potatoes and cabbage. Spread everything into an even layer.

Crack the eggs into a medium bowl. Add the Dijon mustard and whisk until the yolks and whites are fully mixed.

Stir half of the shredded Swiss cheese into the bowl of eggs. Keep the remaining cheese for the top.

Lower the heat under the skillet to medium-low. Pour the egg mixture evenly into the skillet.