Thaw a 17-oz. package of frozen puff-pastry sheets. Preheat oven to 400°F. Chop 2 cups fresh basil, 1 cup fresh cilantro, 1/2 cup shredded Parmesan cheese, 1/4 cup fresh oregano, 1/4 tsp. kosher salt, 1/4 tsp. black pepper, and 2 garlic cloves in a food processor, about 5 pulses. With food processor running, slowly drizzle in 1/4 cup olive oil. Roll out 1 puff-pastry sheet on a lightly floured work surface, maintaining a 9-in. square. Spread half the basil mixture to edges. Fold in both the right side and the left side of the pastry 1 1/2 in. toward the middle. Repeat right and left folds once, meeting in the middle. Fold right side on top of left side. Gently press together. Wrap in foil, and freeze 15 minutes. Repeat with remaining ingredients. Unwrap 1 frozen pastry stack. Cut crosswise into fourteen 1/2-in.-thick slices. Arrange 1 in. apart on two baking sheets lined with parchment paper. Bake sheets one at a time for 9 minutes. Flip pastry; continue baking until golden brown, 4 to 6 minutes. Remove from tray. Repeat with remaining pastry stack. Serve palmeritas warm.
You can store unbaked, frozen slices in an airtight container for up to 3 months.