Preheat oven to 375°F. Bring a large pot of salted water to a boil over high. Add shells; cook, stirring often, over high until al dente, 8 to 10 minutes. Drain shells, running under cold water to help stop cooking.

Stir together ricotta, chicken, goat cheese, Parmesan, egg, basil, garlic, salt, Italian seasoning, black pepper, and 4 ounces (about 1 cup) of the mozzarella in a medium bowl until well combined. Set aside.

Reserve 1 tablespoon sun-dried tomato oil from jar; drain and finely chop tomatoes (about ⅔ cup). Heat reserved oil in a medium saucepan over medium until shimmering. Add chopped tomatoes and onion; cook, stirring frequently, until onions are softened, about 3 minutes. Add marinara sauce, water, and red pepper; bring to a boil over medium, stirring occasionally, then reduce heat to low. Cook, stirring occasionally, over low until flavors meld and slightly reduced, about 15 minutes.

While sauce cooks, place ricotta mixture in a large resealable plastic bag or piping bag; cut a 1-inch hole in corner or tip. Evenly pipe ricotta mixture into cooked shells (about 2 tablespoons each).

Stir cream into sauce until combined. Spread half of sauce over bottom of a 13- x 9-inch baking dish. Arrange shells, filled side up, over sauce; top with remaining sauce.

Cover dish with aluminum foil; bake in preheated oven until shells are tender, sauce is bubbling and filling is heated through, about 30 to 35 minutes. Remove from oven; uncover and sprinkle evenly with remaining 4 ounces (about 1 cup) mozzarella.

Preheat oven to broil with rack about 7 inches from heat source. Broil until cheese is melted and just lightly browned in spots, about 2 to 4 minutes. Garnish with basil. Serve immediately.