Preheat oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.
Combine sun-dried tomato pesto and goat cheese in a small bowl and stir well.
Unroll 1 puff pastry sheet onto the prepared baking sheet. Cut away 2 strips at the bottom of the sheet to form a tree trunk, about 1 inch wide and tall. Cut diagonally to the top of the puff pastry sheet to create a long triangular shape, removing excess pastry on either side.
Spread a thin layer of the pesto mixture over the pastry, all the way to the sides. Sprinkle 1 tablespoon Pecorino-Romano cheese on top.
Roll up the second puff pastry sheet and place at the tip of the triangle. Unroll carefully towards the bottom. Press down lightly and carefully cut away the sides to match the first triangle. Remove excess pastry.
Cut 2/3-inch thick branches into the sides of the triangle from both sides, leaving a space lengthwise in the center all the way from the trunk to the tip. Twist the branches away from you, trying to get in 2 turns on the lower branches. Continue moving up the tree, twisting away from you as you go.
Brush the entire tree with beaten egg. Sprinkle with remaining 1 tablespoon of Pecorino-Romano cheese and dried oregano.
Bake in the preheated oven until deep golden brown, about 20 minutes. Let cool on the baking sheet slightly.