Lightly roll out puff pastry on parchment to smooth out creases. Transfer dough with parchment paper to a baking sheet. Using a knife, lightly score a 1" border around the dough and prick a couple times in the center with form. Brush with egg wash and bake until golden and slightly puffed, about 10 minutes.
Meanwhile, in a medium sized bowl, mix goat cheese and cream. Set aside.
Next, heat 2 tablespoons of olive oil in a medium sized sauce skillet over medium-high heat. Add leeks and cook until they soften. Season with salt, and stir in garlic for 30 seconds.
When puff pastry is done, let cool slightly. If it puffs up too much, prick with a fork to deflate. Spread goat cheese mixture evenly within the 1 inch border. Next, pull leaves off of 2 sprigs of thyme and sprinkle on top. Place potato slices, 1-2 layers deep, on top of goat cheese, drizzling with olive oil and salt in between if doing 2 layers. Top with leeks and black pepper.
Bake until pastry is golden brown and baked through, about 18-20 minutes. Let cool slightly. Sprinkle remaining thyme on top before serving.