Heat oil in a 4 to 5 quart Dutch oven or stock pot over medium until shimmering, then add the bacon. Cook until the fat begins to render and the meaty parts turn golden. Add the onions, celery, and butternut squash, and cook until the onions are translucent and the edges of the squash turn golden. Stir in the broccoli, peppers, and garlic, and continuing heating the vegetables until they're cooked through.

Deglaze the pan with the white wine, mixing well with the vegetables and bacon. Sprinkle the flour, paprika, salt, and black pepper over the vegetables and stir to coat. Add the chicken broth and bring the soup to a light boil. Reduce heat enough to maintain an active simmer for 10 minutes.

Add the gnocchi, increasing heat, if necessary to maintain a simmer. After 5 minutes, reduce heat to low and add the chicken, dairy, cheese, and tarragon, stirring well. Taste for seasonings and adjust as necessary. When the chicken has reheated, ladle into bowls and serve.