Combine water, butter, and salt in a saucepan over medium-high heat. Bring to a simmer; pour in flour all at once and reduce heat to medium. Stir with a wooden spoon or spatula until a dough starts coming together. Cook, scraping up and stirring the dough, for 2 to 3 minutes.
Remove from heat; transfer dough to a mixing bowl. Let cool until no longer hot but still very warm, 5 to 10 minutes. Add egg and season with cayenne and freshly ground black pepper. Whisk vigorously until mixture combines into a very soft, sticky dough. Switch to a spatula and scrape dough into a ball.
Seal dough and spatula with plastic wrap and refrigerate in the bowl until cool, about 1 hour.
In the meantime, season marinara sauce with oregano, red pepper flakes, balsamic vinegar in a small pot over medium heat. Add anchovy fillet. Stir together and bring to a simmer. Let simmer for 10 minutes; turn off heat and let sit until ready to use.
Grate mozzarella cheese over the dough and stir to combine.
Heat oil to 350 degrees F (175 degrees C) in a deep fryer or heavy-duty pan over medium heat. Preheat oven to 200 degrees F (93 degrees C) or any temperature for keeping warm.
Scoop out about 2 tablespoons of dough per puff and form into a football shape using two spoons. Fry puffs, 5 or 6 at a time, in the hot oil until browned, 2 to 3 minutes. Drain on paper towels. Keep puffs warm in low oven while frying the rest. Serve with hot marinara sauce.