Grind the Pizza. Run leftover pizza through a fine meat grinder until you have about 1 cup. Set aside.
Make the Choux Dough. In a saucepan, melt butter. Add water, salt, and ground pizza; whisk to combine.
Stir in flour and cook over medium heat, stirring aggressively, until a smooth paste forms and pulls away from the sides. Mix in paprika.
Remove from heat and add eggs one at a time, mixing vigorously after each addition until glossy and smooth. Transfer dough to a piping bag.
Pipe and Cook the Gnocchi. Bring heavily salted water to a gentle simmer, not a rolling boil.
Pipe dough directly into the water, cutting each piece with a knife or offset spatula.
Cook until gnocchi float and are cooked through. Remove with a spider and transfer to an oiled sheet tray.
Cool completely, then refrigerate overnight for best texture.
Crisp the Pepperoni and Gnocchi. In a skillet over medium heat, render julienned pepperoni in olive oil.
Add cooked gnocchi and sauté until deeply golden and crispy. Season with salt.
Make the Tomato Basil Topping. Mix basil, pickled peppers, tomato conserva or cherry tomatoes, salt, black pepper, and chili oil in a bowl.
Finish and Serve. Spoon the tomato basil mixture over hot crispy gnocchi.
Top with torn burrata, flaky salt, black pepper, and an extra drizzle of chili oil. Serve immediately.