Place the unsalted butter in a small saucepan over medium heat. Stir frequently as it melts, then begins to foam and brown. Once the butter turns a golden brown color and smells nutty (about 5–7 minutes), immediately remove it from the heat. Pour into a heatproof bowl and let it cool slightly.
Combine the granulated sugar, ground cinnamon, and salt in a mixing bowl. Stir together until evenly mixed. Add the vanilla extract to the browned butter and whisk to combine.
Unfold the thawed puff pastry sheets on a lightly floured surface. If the sheets are too stiff, gently roll them out with a rolling pin just enough to smooth any creases.
Using a pastry brush, generously coat the entire surface of the puff pastry with the brown butter mixture. Then evenly sprinkle the cinnamon sugar mixture over the buttered surface, pressing lightly to help it stick.
Starting from one long edge, carefully roll the pastry tightly towards the center. Repeat with the opposite edge so both sides meet in the middle, forming a double roll. Gently press the rolled pastry together and wrap tightly in plastic wrap. Refrigerate for at least 30 minutes to chill and firm up.
Preheat oven to 400°F (205°C). Remove the rolled pastry from the fridge and unwrap it. Using a sharp knife, slice the roll into 1/4-inch thick pieces. Place each slice cut-side up on a parchment-lined baking sheet, leaving space between each palmier.
Bake for 12–15 minutes, or until the palmiers puff up and turn a deep golden brown with caramelized sugar edges. Keep an eye on them to avoid burning. Transfer to a cooling rack and let cool completely before serving.