Put the softened cheese and butter in the bowl of a stand mixer. Using the paddle attachment, beat until the mixture has the consistency of whipped cream, 15 to 30 minutes.
Sift 1 cup of the flour together with the salt, garlic powder, black pepper and cayenne in a large bowl. Stir in the thyme. Gradually add the seasoned flour to the cheese mixture by large spoonfuls, beating well after each addition. Add the remaining 1 cup unseasoned flour incrementally until the dough is somewhat stiff but still soft enough to be pushed through a cookie press (you may not need to add all the flour).
Lightly spray 2 cookie sheets with cooking spray. Put a portion of the dough into a cookie press fitted with the star tip and press the dough onto the cookie sheets in long strips that run the length of the pans (see Cook's Note). Repeat with the remaining dough until the cookie sheets are full. Bake, flipping halfway through, until the straws are golden brown and crisp, about 20 minutes.
With your hands or a sharp knife, break or cut the strips into 3-inch lengths. Use a flat, thin spatula or egg turner to remove the cheese straws from the pan. Allow them to cool on a wire rack. Once they are completely cool, serve or store in a tightly covered container.