Pat chicken dry; season both sides with dried basil, pepper, and 1 tsp. salt. In a large Dutch oven or pot over medium heat, heat 2 tbsp. oil. Cook chicken, turning occasionally, until an instant-read thermometer inserted into thickest part registers 160°, 8-10 minutes per side. Transfer to a plate.

In same pot over medium heat, cook shallots in remaining 1 tbsp. oil, stirring, 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 4 minutes. Stir in wine, scraping bits from bottom of pot, and cook, stirring occasionally, until reduced by two-thirds. Stir in tomatoes. Tie 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 tbsp. salt and simmer until slightly reduced, about 10 minutes.

Meanwhile, in a small saucepan over high heat, bring vinegar and sugar to a boil, stirring to dissolve sugar. Reduce heat to medium-low and simmer until thickened, about 5 minutes.

Shred chicken into bite-size pieces. Discard basil stems. Add gnocchi and remaining 1 tsp. salt and cook, stirring, until gnocchi is cooked through, 4 to 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.