How to Make It
In a food processor, pulse the flour, tomato paste, sugar, cheese, thyme, salt, cayenne and black pepper until the tomato paste is evenly distributed. Add the butter and pulse until the dough starts to come together. Scrape the dough onto a large sheet of plastic wrap and roll it into an 8-inch log, 1 1/2 inches in diameter. Wrap the dough in the plastic and refrigerate until well chilled, at least 1 hour.
In a medium bowl, combine all of the ingredients and season with salt and black pepper.
Preheat the oven to 350° and line 2 baking sheets with parchment paper. Slice the shortbread log into 1/4-inch-thick rounds and arrange on the prepared baking sheets. Bake the shortbreads in the upper third and center of the oven for about 18 minutes, until the edges are lightly browned; rotate the baking sheets from top to bottom and front to back halfway through baking. Transfer the shortbreads to a wire rack to cool completely. Serve with the olive gremolata.