CHICKEN: Pat the chicken dry. In a small bowl, combine Italian seasoning, 1 teaspoon salt, and pepper, sprinkle over both sides of the chicken.
Heat 2 tablespoons olive oil in a large Dutch oven or pot over medium heat. Add chicken, turning occasionally, until an instant-read thermometer inserted into thickest part registers 160 degrees, about 4-5 minutes per side. Transfer chicken to a plate and loosely cover with foil.
SOUP: Add 1 tablespoon olive oil in the same pot over medium heat, cook shallots until softened, about 2 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute.
Add tomato paste and cook, stirring, until paste darkens, about 3-4 minutes. Stir in wine, scraping bits from bottom of pot, and cook, stirring occasionally, until reduced by two-thirds.
Stir in tomatoes. Tie the basil stems together and add to pot. Cook, breaking down tomatoes with a wooden spoon, until thickened and jammy, about 10 minutes.
Add broth and 1 teaspoon salt, simmer until slightly reduced, about 10 minutes.
Meanwhile, shred the chicken into bite-size pieces. Discard basil stems.
ENOCCHI: Add gnocchi and remaining 1 tsp. salt. Cook, stirring, until gnocchi is cooked through, about 4-6 minutes.
Return chicken to pot along with half of torn basil and stir to heat through.
Divide soup among bowls. Top with mozzarella, croutons, remaining torn basil, and a drizzle of balsamic glaze.