Whisk the flour, cheese powder, and salt together in a small bowl.
Place 3 tablespoons of the butter and the starter in the work bowl of a stand mixer fitted with the paddle attachment, and work on low speed until creamy and homogeneous, about 1 minute. Stop the mixer and sift the flour mixture over the creamed starter, return the mixer to low, and work until a soft dough forms, about 1 minute more.
Turn the dough out onto a lightly floured surface and divide into 4 equal pieces. Press each piece into a 1/4-inch-thick rectangle and wrap in individual pieces of plastic wrap. Refrigerate between 8 and 24 hours (see Cook's Note).
Place two racks nearest the center of the oven and heat to 350ºF. Line two half-sheet pans with parchment paper.
Retrieve one piece of dough from the refrigerator and dust both the dough and your pasta maker with flour. (Continue to dust the pasta roller and the dough as needed to prevent sticking.) Starting on the widest setting, typically labeled "1," feed the dough through the roller. Then fold the dough into thirds, as you would a letter, turn 90 degrees and run it through again. Fold the dough into thirds again, turn, and pass it through the roller.
Adjust your pasta maker to the next setting and roll the dough through 3 times without folding. Repeat this step 3 times more on both the 3rd and 4th settings. Transfer the dough to one of the prepared half-sheet pans and repeat the process with the remaining dough, arranging two sheets of dough per pan.
Melt the remaining 1 tablespoon of butter and brush onto the dough. Sprinkle with a generous pinch of kosher salt. Dock the dough thoroughly with a fork and cut into 1-inch squares with a pizza wheel.
Bake for 10 minutes, then turn the pans 180 degrees, and rotate from top to bottom. Continue baking until the crackers just begin to brown around the edges, 8 to 10 minutes more.
Remove from the oven and immediately transfer the crackers on the parchment to a wire rack to cool completely before serving, about 20 minutes.
Lid tightly and store at room temperature.