In a mixing bowl, combine the unbleached all-purpose flour, whole-wheat flour, baking powder, fine salt, and sweet paprika.
Add the grated cheese and softened butter to the dry ingredients, mixing until the mixture resembles coarse crumbs.
Gradually incorporate the ice water, one tablespoon at a time, until the dough comes together. You may need 2 or 3 tablespoons.
Once the dough is formed, wrap it in plastic wrap and refrigerate for about 30 minutes to firm up.
Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
On a lightly floured surface, roll out the dough to about ⅛ inch thick.
Cut the dough into squares or desired shapes and place them on the prepared baking sheet.
Sprinkle the tops with kosher salt before baking.
Bake in the preheated oven for 15-20 minutes, or until the Cheez-Its are golden and crispy.
Allow them to cool on a wire rack before serving.