Heat oven to 400° F. Lightly coat 2 cookie sheets with vegetable cooking spray. In a medium saucepan, combine 1 cup of water with the butter, oil, salt, and cayenne. Bring to a boil. Remove from heat and add the flour all at once. Stir with a wooden spoon until the ingredients form a smooth paste.
With a wooden spoon, stir in 1 egg until completely blended. Repeat with the remaining 3 eggs. Stir in the Cheddar and chives. Using a teaspoon, drop balls of batter onto the cookie sheets. Bake 20 to 25 minutes or until the puffs are firm and well browned. Serve warm or at room temperature.
To keep for up to 2 days, cut a small slit in the side of each puff immediately after removal from the oven. Cool completely. Store in an airtight container.