Heat oil in a large pot over medium-high; season chicken with salt and pepper and cook 6-8 minutes until golden brown and cooked through. Set aside.
Melt butter in same pot over medium; cook onion about 5 minutes until softened, then garlic another minute until fragrant.
Sprinkle flour over onions; stir constantly 2 minutes to form a roux.
Gradually whisk in broth, smoothing lumps; bring to a gentle simmer.
Reduce heat to low; stir in cream, chicken, basil, Parmesan, and sun-dried tomatoes.
Simmer gently 15 minutes, stirring occasionally, until thickened and flavors meld; adjust salt and pepper.
Serve hot topped with crispy prosciutto and extra fresh basil, with crusty bread for dipping.