Drain olives well and pat dry with a paper towel.
In a medium bowl, stir together butter and cheddar with a wooden spoon until well combined.
Add flour, paprika, salt, pepper, and cayenne pepper. Stir once more and finish mixing with your hands, if needed. Dough should be moist but dense and easy to form into a ball. If needed, add more flour by the tablespoon until desired consistency is reached.
Chill dough for 30 minutes.
Pinch off a small piece of chilled dough, then flatten into a thin oval shape. Wrap dough around the olive, covering it completely, then pinch dough closed.
Place on a baking sheet. Continue with remaining dough; you will have some leftover olives.
Optional: If you'd like more of a ball shape after baking rather than a wafer, chill at least 1 hour and up to overnight.
When ready to bake, preheat oven to 400°F. Line a rimmed baking sheet with parchment paper.
Transfer olives to prepared baking sheet, leaving space around each olive for air to circulate. Bake until lightly golden, 15-18 minutes.
Let cool a few minutes before serving. Enjoy!
