Heat the oven to 400°F with a rack in the middle. Brush a 9-inch square baking dish with 1 tablespoon of the olive oil.

Spread the torn bread on a sheet pan. Put the pan in the oven and toast the bread for 8 to 10 minutes, until the pieces feel dry on the surface and lightly crisp at the edges.

Heat the remaining 1 tablespoon olive oil in a large skillet over medium heat.

Add the leeks and 1/2 teaspoon of the salt to the skillet. Cook for 6 to 8 minutes, stirring often, until the leeks are very soft and lightly golden.

Add the dried thyme to the skillet. Stir for about 30 seconds, just until it smells fragrant.

Add the spinach to the skillet a few handfuls at a time. Cook for 2 to 3 minutes, stirring, until the spinach wilts and most of the extra moisture cooks away.

Put the drained butter beans into the skillet. Stir gently for 1 to 2 minutes to warm them through, then take the skillet off the heat.

Put the toasted bread into the prepared baking dish and spread it into an even layer.

Spoon the leek, spinach, and butter bean mixture over the bread in the baking dish.

Sprinkle about 1 cup of the grated cheddar over the mixture in the baking dish. Reserve the rest of the cheddar for the top.

Crack the eggs into a large bowl.

Add the milk, Dijon mustard, remaining 1/2 teaspoon salt, and the black pepper to the bowl. Whisk until the mixture is smooth and evenly combined.