Place the thyme, orange zest and sugar in the bowl of a stand mixer fitted with the paddle attachment. Use your hands to mix the thyme and orange zest into the sugar. This ensures the thyme and orange zest end up evenly incorporated throughout the cookie dough.
Beat the butter with the orange thyme sugar at low to medium speed until creamy and well incorporated.
Add the egg yolk and beat until just combined, about 1 minute.
Slowly drizzle in the olive oil and beat until smooth and well incorporated. The dough will have a lovely sheen at this point.
Add the salt and flour and beat until just incorporated.
Turn the cookie dough out onto a work surface or piece of parchment paper and press it until it just comes together. Divide the dough in half and press each half into a disk. (or make a log out of one or both of the pieces of dough to make slice and bake cookies).
Roll out each disk between 2 sheets of wax parchment paper to about ¼ inch thick. Slide the wax paper–covered disks onto a baking sheet and freeze for at least 1 hour, until very firm.
Preheat the oven to 350°F and line 2 baking sheets with parchment paper (if making all the cookies at once).
Working with one piece of cookie dough at a time, peel off the top sheet of wax paper. Using a 1 ½-inch round cookie cutter, stamp out the cookies as close together as possible.
NOTE: You can reuse the dough remnants but you will need to refreeze the dough to work with it.
Arrange the cookies 1-inch apart on the prepared baking sheets. Bake the cookies for 15 - 18 minutes, until they are lightly golden; shift the baking sheets from top to bottom and front to back halfway through. If baking two sheets at a time, your cook time might be a tad longer. Look for the color cues of a slight golden edge. (If your dough is a little thinner than ¼ inch or you use a smaller cookie cutter, reduce the cooking time - start checking them at 10 minutes).
Let the cookies cool on the baking sheets for 3 minutes, then transfer them to a wire rack to cool completely.