Step 1Combine the butter, mustard, salt, cayenne and black pepper in a medium heavy saucepan over medium heat. Add 1 cup water and bring to a boil. Add the flour all at once and beat quickly with a wooden spoon until the ingredients come together into a ball that pulls away from the sides of the pan.
Step 2Remove the pan from the heat and immediately beat in two of the eggs, beating until they are completely incorporated. Return to the heat briefly while continuing to beat, then remove from the heat and beat in the remaining eggs, beating until the dough is very smooth. Return to the heat and cook, stirring, for 1 to 2 minutes, until the dough is shiny and smooth. Cool slightly, then beat in the cheese.
Step 3Pour about 3 inches of oil into a deep fryer and heat to 375 degrees. Working in small batches, drop the dough by the rounded teaspoon into the hot oil. Fry, turning once, until each beignet is puffed and evenly browned on all sides, 3 1/2 to 4 minutes. Drain on paper towels before serving.