Preheat the oven to 400°F. Line a baking sheet with parchment paper. Lightly dust a clean surface with flour.
Flour a rolling pin and roll out the puff pastry slightly to even out the surface. Aim for a rectangle about 10x12 inches.
Evenly spread the vegan pesto over the surface of the puff pastry, leaving about ½ inch border on all sides. Scatter the finely chopped sundried tomatoes on top of the pesto. Add a pinch of salt and freshly ground black pepper for seasoning.
Starting with one of the longer edges, roll the pastry tightly toward the center until you reach the middle.
Repeat with the opposite edge so the two rolls meet in the center.
Wrap the rolled pastry in plastic wrap and refrigerate for 15–20 minutes to firm up.
Remove the pastry from the fridge and slice it into ½-inch thick slices using a sharp knife.
Lay the slices flat on the prepared baking sheet, leaving about 1 inch between each palmier.
Bake for 15–18 minutes or until the palmiers are puffed and golden brown. Rotate the baking sheet halfway through baking for even cooking.
Allow the palmiers to cool slightly on the baking sheet before transferring to a wire rack. Serve warm or at room temperature.