Start the skillet: Set a 10-inch skillet with a lid over medium heat.

Warm the oil: Add the olive oil to the skillet and let it warm for about 30 seconds.

Cook the onion: Add the chopped onion and 1/2 teaspoon of the kosher salt to the skillet. Cook, stirring often, until the onion looks soft and lightly golden at the edges, 4 to 5 minutes.

Add the broccoli: Add the broccoli, black pepper, and red pepper flakes to the skillet. Stir until the broccoli is coated in the oil and onions.

Steam until tender: Pour 1/4 cup of the water into the skillet. Cover with the lid and cook until the broccoli is bright green and the thicker pieces can be pierced easily with a fork, 3 to 4 minutes.

Add the beans: Uncover the skillet and stir in the drained butter beans.

Loosen the mixture: Pour the remaining 1/2 cup water into the skillet. Stir gently so the beans stay mostly whole.

Mash a few beans: Use the back of a wooden spoon to lightly mash about one-third of the beans right in the skillet. This helps thicken the base for the eggs.

Simmer the base: Cook the bean and broccoli mixture uncovered over medium heat until it looks slightly thickened and no longer watery, 3 to 4 minutes. Stir once or twice so the bottom does not stick.

Season with lemon and dill: Stir in about half of the lemon zest, 1 tablespoon of the lemon juice, and half of the chopped dill. Taste the mixture and add more salt if it needs it.

Make room for the eggs: Use a spoon to make 4 shallow wells in the bean mixture. Space them out so each egg has room to cook.

Add the eggs: Crack 1 egg into each well in the skillet. Sprinkle a small pinch of salt over the eggs.