Croque Madame with Mushrooms

Preheat the broiler with a rack about 6 inches from the heat. Slice the cremini mushrooms and grate the Cabot cheddar if not already prepared.

In a medium saucepan, melt 3 tablespoons butter over medium heat. Whisk in the flour for 1 minute, then slowly whisk in the whole milk and cook 3 to 4 minutes, until thick enough to coat a spoon.

Whisk in the Dijon mustard, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Remove from the heat and stir in 4 ounces Cabot cheddar until the béchamel is smooth and cheesy.

Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the mushrooms, 1/4 teaspoon salt, and 1/4 teaspoon pepper, then cook 8 to 10 minutes, stirring occasionally, until browned and the pan is mostly dry.

Melt the remaining 1 tablespoon butter and brush it lightly over the bread slices. Arrange on a rimmed baking sheet and broil 1 to 2 minutes per side, until lightly toasted.

Spread 4 toast slices with about 2 tablespoons béchamel each, then divide the sautéed mushrooms over them. Top with the remaining toast slices, spoon the remaining béchamel over the sandwiches, and sprinkle with the remaining 2 ounces cheddar.

Wipe out the skillet and heat the remaining 1 tablespoon olive oil over medium-low heat. Crack in the eggs, season with the remaining 1/4 teaspoon salt, cover, and cook 4 to 6 minutes, until the whites are fully set and the yolks are firm and fully cooked.