Prepare the vegetables and aromatics: dice the onion, carrot and celery into small, even pieces; mince the garlic, slice the sun-dried tomatoes, and chiffonade the basil.
Warm butter and the sun-dried tomato oil in a large Dutch oven over medium heat. Add the onion, carrot and celery and cook, stirring every couple minutes, until the vegetables are softened and starting to turn translucent, about 6–8 minutes.
Pour in the white wine and scrape any browned bits from the bottom of the pan; let it reduce by half (about 1–2 minutes). Stir in the tomato paste and sun-dried tomatoes and cook until the paste darkens slightly.
Add the minced garlic, garlic-herb seasoning, herbes de Provence, red pepper flakes, salt and pepper. Cook 30–45 seconds more, just until the garlic becomes fragrant.
Pour in the chicken broth and heavy cream and bring the mixture to a gentle simmer. Once simmering, add the gnocchi and the cooked chicken. Simmer until the gnocchi are tender and plump, about 3–5 minutes (they will become noticeably larger and soft).
Remove the pot from the heat. Whisk in the softened cream cheese in batches until fully incorporated and smooth. Stir in the Parmesan and balsamic vinegar, tasting and adjusting salt and pepper as needed.
Fold in the baby spinach and most of the basil; the residual heat will wilt the greens while keeping their color bright. Let the soup sit off the heat for 5 minutes to thicken slightly before ladling into bowls and garnishing with the remaining basil.