Set a rack in the upper middle of the oven and heat the oven to 425°F.

Crack the eggs into a medium bowl. Add the milk, 1/2 teaspoon of the salt, and 1/4 teaspoon of the black pepper.

Whisk the egg mixture until the yolks and whites are fully blended, then set the bowl on the counter near the stove.

Set a 10-inch oven-safe skillet over medium heat. Add the olive oil and butter to the skillet.

When the butter melts, put the torn bread into the skillet. Stir the bread for 3 to 4 minutes, until the edges look lightly golden and a little crisp.

Transfer the toasted bread from the skillet to a plate.

Keep the skillet over medium heat and add the chopped onion. Sprinkle in 1/4 teaspoon of the salt.

Cook the onion for 3 minutes, stirring now and then, until it softens and looks glossy.

Add the broccoli florets to the skillet. Stir them with the onion for 1 minute so they are coated in the fat in the pan.

Pour 2 tablespoons of water into the skillet. Cover the skillet with a lid or a baking sheet and cook for 2 minutes to help the broccoli soften.

Uncover the skillet and stir the broccoli. Cook for 2 more minutes, until the broccoli is bright green and just tender when you bite a small piece.

Return the toasted bread to the skillet and spread it into an even layer among the broccoli and onion.