Preheat oven to 425 degrees F. Lightly coat 2 baking sheets with cooking spray or line with parchment paper.Advertisement
Combine water, butter, oil, salt and cayenne in a medium saucepan; bring to a boil over medium heat. Remove from heat and add flour. Stir with a wooden spoon until it forms a smooth paste. Return the mixture to low heat and cook, stirring, for about 3 minutes, to dry the paste slightly. Remove from heat and cool for 2 minutes.
Lightly whisk together whole eggs and 3 egg whites. Beat one quarter of the egg mixture into the flour paste with a wooden spoon until absorbed. Repeat with 3 more additions of the egg mixture, until the paste is smooth and glossy. Stir in 3/4 cup of the cheese.
Drop 1-inch mounds 1 1/2 inches apart on the prepared baking sheets, using a pastry bag or spoon. Lightly beat the remaining egg white and brush it over each puff. Sprinkle with the remaining 2 tablespoons cheese. Bake, one sheet at a time, until the puffs are firm and well browned, 20 to 25 minutes. Serve warm.
Make Ahead Tip: Freeze, well-wrapped, for up to 2 months. Reheat for 10 minutes at 350°F.