Arrange a rack in the middle of the oven and heat the oven to 300°F. Coat an 8x8-inch baking dish generously with cooking spray or butter.

Whisk 8 large eggs in a large bowl until blended. Add ½ cup milk and ½ teaspoon kosher salt and whisk until combined. Pour into the baking dish.

Pull the oven rack halfway out of the oven. Place a rimmed baking sheet on the oven rack. Pour in 3 cups hot tap water. Set the baking dish of eggs in the baking sheet and carefully push the oven rack back in.

Bake until just set in the middle, about 30 minutes. Meanwhile, grate 4 ounces Gruyère or sharp cheddar cheese on the large holes of a box grater (about 1 cup). Thinly slice 3 ounces peppadew peppers (about ½ cup). Stir together 2 tablespoons Dijon mustard and 2 tablespoons mayonnaise in a small bowl.

Carefully lift the baking dish out of the baking sheet to a wire rack and let cool for 5 minutes. Meanwhile, remove the baking sheet from the oven, pour off any remaining water, and wipe dry. Switch the oven to broil.

Run a thin knife around the edges of the soufflé, then flip it out onto a cutting board. Cut into 4 squares. Using a flat spatula, carefully transfer the squares to one side of the baking sheet. Sprinkle with the grated cheese. Split 4 brioche sandwich rolls and place them cut-side up on the other side of the baking sheet. Broil until the cheese is melted and the buns are lightly toasted, 30 seconds to 1 ½ minutes.

Spread the dijonaise onto the cut sides of each roll. Top each roll bottom with an egg soufflé square. Top with 1 packed cup baby arugula and the sliced peppadew peppers. Close the sandwiches with the roll tops and serve.