In a large mixing bowl, whisk together the flour, salt, and paprika (if using).

Add the cold, cubed butter to the flour mixture. Using your fingertips or a pastry blender, cut the butter into the flour until it resembles coarse crumbs with some pea-sized pieces remaining.

Stir in the sourdough discard and finely grated sharp cheddar cheese. Mix until a shaggy dough forms. If the dough is too dry to come together, add ice water one tablespoon at a time until it just forms a cohesive ball. Be careful not to overmix.

Flatten the dough into a disc, wrap it tightly in plastic wrap, and refrigerate for at least 30 minutes, or up to 2 days.

Preheat oven to 350°F (175°C). Line two large baking sheets with parchment paper.

Divide the chilled dough in half. On a lightly floured surface or between two sheets of parchment paper, roll out one half of the dough very thinly, about 1/16 to 1/8-inch thick. The thinner the dough, the crisper your crackers will be.

Using a pizza cutter or a sharp knife, cut the dough into 1-inch squares. Carefully transfer the squares to the prepared baking sheets, leaving a little space between them.

Using a skewer, chopstick, or the blunt end of a small utensil, poke a hole in the center of each cracker square. This helps prevent them from puffing up too much during baking.

Repeat with the remaining dough. You can re-roll scraps once or twice.

Bake for 12-18 minutes, rotating the baking sheets halfway through, until the crackers are golden brown and crispy. The edges might be a bit darker. Keep a close eye on them, as they can go from perfectly golden to burnt quickly.

Remove from the oven and let cool completely on the baking sheets. They will crisp up further as they cool.