Thaw the puff pastry from frozen for about 40 minutes before commencing. Preheat the oven to 425°F.
Bring a small pot of salted water to a boil. Thinly slice the russet potato using a mandoline or very sharp knife, then drop the slices into the boiling water and cook for 1-2 minutes. Drain the water, reserving the "cooked" potato slices.
Clean the mushrooms with a cloth rag, then slice them into about ⅛-to-¼ inch thick pieces. Place a large saute pan over medium-high heat, add a drizzle of olive oil, and once the oil shimmers, add the sliced mushrooms. Season with salt and pepper and cook for just 2-3 minutes. Remove from the heat and set aside.
Place a silpat baking mat or parchment paper on a quarter or half sheet pan. Spread the defrosted puff pastry out onto the prepared sheet pan. Roll the pastry out on the silpat to form a rectangle of roughly 10x15 inches. Poke the puff pastry all over with a fork to prevent the dough from rising. Place in the oven for 10-15 minutes, until lightly toasted.
In a food processor, blend the goat cheese with the creme fraiche and egg until the mixture reaches a spreadable consistency. Spread the cheese mixture over the parbaked puff pastry, leaving a small rim around the edges of the pastry uncovered, then cover the soft cheese spread with the shredded cheese. Decorate the top with the prepared potato slices, another drizzle of olive oil, a dusting of the grated Parmesan cheese, the chopped fresh rosemary, and a sprinkle of Kosher salt.
Place the puff pastry in the preheated oven and bake for about 40-50 minutes. While it bakes, prepare the arugula. Rinse to clean, pat or spin dry, then toss with kosher salt, olive oil, and lemon juice to taste.
Lift the pizza a little to look at the bottom, it should be golden brown when close to done. At this point, add the prepared mushrooms and the seasoned arugula, then return to the oven for another 2-3 minutes, until the pizza is done. Remove from the oven, slice into 4-8 segments and serve.
