Beat Cheddar cheese, butter, half-and-half, and salt at medium speed with a heavy-duty electric stand mixer until blended. Gradually add flour, Parmigiano-Reggiano cheese, and basil, beating just until combined.
Turn dough out onto a well-floured surface. Divide into 4 equal portions, and flatten each into a disk; roll to 1/8-inch thickness. Cut with a 1 1/2-inch star-shaped cutter. Place 1 inch apart on parchment paper-lined baking sheets. Bake 11 to 14 minutes or until golden; cool on baking sheets on wire racks 30 minutes.