Set a 12-inch skillet with a lid on the stove. Put the potatoes in the skillet with 3/4 cup water, 1 tablespoon of the olive oil, and 1/2 teaspoon of the salt.

Cover the skillet and set it over medium-high heat. Cook for 10 to 12 minutes, stirring once halfway through, until the potatoes are just tender when pierced with a knife.

Uncover the skillet. If more than a spoonful of water is left, cook for 1 to 2 minutes more until the pan is nearly dry.

Add the shallot and the remaining 1 tablespoon olive oil to the skillet. Cook over medium heat for 3 minutes, stirring often, until the shallot softens.

Add the garlic to the skillet. Cook for 30 seconds, stirring, until it smells fragrant.

Put the green beans into the skillet. Add 1/4 teaspoon of the salt and stir so the beans touch the hot pan.

Cover the skillet again and cook for 3 to 4 minutes, stirring once, until the green beans are bright green and just tender.

In a small bowl, stir together the mustard, vinegar, 2 tablespoons water, the black pepper, and the remaining 1/4 teaspoon salt.

Pour the mustard mixture into the skillet. Stir gently until the potatoes and green beans look glossy and lightly coated.

Cook uncovered for 1 minute over medium heat so the dressing warms and clings to the vegetables.

Lower the heat to medium-low. Use the back of a spoon to make 4 shallow spaces in the vegetables.

Crack 1 egg into each space in the skillet. Keep the eggs separate so they cook evenly.