Make the dough: Bring 4 cups water to a boil in a 6-qt. saucepan. Add pasta and reduce heat to medium; cook, stirring occasionally, until pasta is very soft, about 20 minutes. Let pasta cool in water; drain completely and transfer to a food processor. Add salt and purée into a smooth, sticky dough. Divide dough between 2 sheets of greased parchment paper. Top each with another sheet of greased parchment paper and flatten; chill 30 minutes.
Make the filling: Heat 1 tbsp. oil in a 12" skillet over medium-high. Cook sausage, stirring and breaking up into small pieces, until browned, 4–6 minutes. Transfer sausage to a colander set over a bowl; set aside. Add remaining oil to skillet. Cook garlic until golden, 1–2 minutes. Stir in broccoli rabe, salt, and 1⁄4 cup water; boil. Reduce heat to medium; cook, covered, until broccoli rabe is tender, 6–8 minutes. Transfer to colander with sausage and let filling cool. Transfer filling to a bowl; stir in parmesan and peppers.
Assemble the ravioli: Working with half the dough and using a rolling pin, roll dough 1⁄6" thick. Remove top sheet of parchment paper; using a 4 1⁄2" round cutter, cut out 10 circles, gathering and rerolling scraps. Place 1 slice mozzarella over each circle and top with 2 tbsp. filling. Fold circle in half and pinch edges to seal in filling.
Fry the ravioli: Heat 2" canola oil in a 6-qt. saucepan until a deep-fry thermometer reads 350°. Place flour, eggs, and bread crumbs in separate shallow dishes. Working with 1 ravioli at a time, dredge in flour, dip in eggs, and coat in bread crumbs; fry until golden and crisp, 2–3 minutes. Transfer ravioli to paper towels to drain; serve hot.