Heat the oven to 425°F with a rack in the middle position. Line a rimmed baking sheet with kitchen parchment. In a small bowl, beat the egg yolk with 1 teaspoon water until well combined. In a medium bowl, combine all cheeses, the thyme, lemon zest and ½ teaspoon each salt and pepper. If you are using DuFour puff pastry, unfold the dough onto a cutting board, then cut it in half crosswise. On a lightly floured counter, roll each piece into a 9-by-12-inch rectangle. If you are using Pepperidge Farm puff pastry, on a lightly floured counter, roll each sheet into a 9-by-12-inch rectangle.
Place 1 sheet on the prepared baking sheet. Distribute the cheese mixture over it, leaving a ½-inch border around all edges. Brush the edges with egg yolk, then lay the second sheet on top; firmly press the edges to seal. Brush the surface with egg yolk, then, using a fork, poke holes at 1-inch intervals into the top. Bake until puffed and golden brown, 20 to 25 minutes.
Let cool on the baking sheet on a wire rack for 10 minutes. Slide the parchment with khachapuri directly onto the wire rack and cool for at least 10 minutes. Serve warm or at room temperature.
