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In the bowl of an electric mixer fitted with the paddle attachment, cream butter and sugar on high speed until light and fluffy, about 5 minutes. Add egg, vanilla, and salt; mix on medium speed until combined. With mixer on low speed, add flour in 2 batches, mixing just until incorporated. Add enough food coloring until you reach desired shade of black.
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Divide dough into two equal portions. Place each portion on a piece of plastic wrap; flatten into disks. Wrap in plastic wrap; refrigerate until firm, at least 2 hours or up to 1 week.
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Preheat the oven to 350 degrees. Line baking sheets with nonstick baking mats or parchment paper. Lightly dust work surface with flour. Roll out one portion of dough to 1/8-inch thick. Using a 3-inch cat paw cookie cutter, cut out cookies.
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Transfer cut dough to prepared baking sheets. Insert an almond "nail" into the toe of each paw. If dough becomes too soft, place in the refrigerator for a few minutes firm. Bake cookies until golden around the edges and slightly firm to the touch, 12 to 15 minutes. Let cool on baking sheet 1 minute; transfer to wire racks to cool completely. Repeat process with remaining dough.
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Transfer half of the royal icing to a plastic squeeze bottle. Add enough black food coloring to remaining icing to make it grey. Transfer to a plastic squeeze bottle. To make the paw pads, squeeze 1 large circle of grey or white icing, about 1 1/4 inches in diameter, on the bottom third of the paw. Squeeze smaller circles, about 1/2 inch in diameter, in the center of each toe. Repeat with remaining cookies and white and grey icing. Allow icing to set completely before serving.