Chef John's Cheesy Crackers

Ingredients

• 1/2 teaspoon vegetable oil
• 2 tablespoons unsalted butter at room temperature
• 3/4 cup lightly packed shredded sharp Cheddar cheese
• 1/3 cup lightly packed freshly shredded Parmesan cheese
• 1/2 teaspoon paprika
• 1 pinch cayenne pepper (or to taste)
• 1/4 teaspoon salt
• 1/2 cup all-purpose flour
• 1 tablespoon cold water (or as needed)