Add the cottage cheese to a small blender and blend for about 60 seconds until completely smooth with no visible curds. It should look like thick sour cream. Smooth cottage cheese keeps the dough consistent and the texture even.
Add the blended cottage cheese, grated cheddar, softened butter, and sourdough to a food processor. Pulse 8 to 10 times until combined. Add the flour, kosher salt, garlic powder, mustard powder, and smoked paprika. Pulse just until the dough comes together into a shaggy ball. Do not overmix or the straws will be tough.
Turn the dough out and divide it in half, then cut each half in half again so you have four equal portions. Working in smaller portions gives you better control when rolling. Press each into a flat disc, wrap tightly, and refrigerate for at least 45 minutes or up to 48 hours.
For a standard oven, preheat to 375 degrees F. For a convection oven, preheat to 350 degrees F since the fan circulates heat. Line two baking sheets with parchment paper.
Remove one dough portion and let it rest at room temperature for 15 minutes before rolling. This relaxes the gluten and makes the dough easier to work with. Keep the other portions cold.
Flour the dough well. Roll it out as thin as you possibly can, about 1/16 inch, since the thinness of the dough is what gives you a crispy straw. You can roll by hand between two sheets of parchment, or run it through a pasta machine starting at setting 1, then 3, then 5, flouring lightly before each pass.
Using a pizza cutter or pastry wheel, cut the rolled dough into 1/2 inch wide strips. Pick up each strip and twist it a few times into a spiral, then lay it on the parchment lined sheet. The length is flexible, so cut them shorter or longer to your liking. Sprinkle with flaky sea salt.
Bake for 15 to 20 minutes until golden and crisp all the way through. Start checking at 12-15 minutes. Because the twisted shape is thicker than a flat cracker, watch for them to turn deeply golden and firm. Ovens that run hot will finish closer to 15, while cooler ovens may need the full 20.
Turn the oven off and leave the straws inside for 5 more minutes to pull out the last of the moisture. Watch closely if using convection since the retained heat is more intense.
Transfer to a wire rack and cool completely. The straws continue to crisp as they cool. Do not store them warm or they will steam soft.