Add the softened butter, grated cheddar, flour, smoked paprika, and salt to a food processor. Pulse everything together until it forms a soft, slightly crumbly dough that holds together when pressed between your fingers.
Turn the dough out onto a sheet of plastic wrap and shape it into a log about 2 inches thick. Roll the plastic tightly around it, twisting the ends closed so the shape stays firm and even.
Place the wrapped log in the refrigerator for at least 30 minutes. This step firms up the butter again, which makes slicing so much easier and keeps the crackers from spreading too much in the oven.
Once chilled, unwrap the log and slice it into rounds about a quarter-inch thick. Lay the slices on a parchment-lined baking sheet, leaving a little space between each one.
Sprinkle the tops lightly with flaky sea salt, then bake at 350°F for 12 to 15 minutes, until the edges turn golden brown. Let the crackers cool completely on a wire rack before serving, since they crisp up further as they sit.