Pat chicken dry and generously salt and pepper both sides.
In a large pot, heat 1 tablespoon of the oil over medium-high heat. Cook the chicken until well-browned, 3 to 4 minutes, flip and repeat. Transfer chicken to a plate and set aside.
Turn heat down to medium and to the same pot, add remaining 1 tablespoon of the olive oil. Add garlic, onion, celery, and carrots, and cook, stirring occasionally, until the vegetables begin to soften, 5 to 8 minutes.
Stir in the white wine, using a wooden spoon to scrape any cooked-on bits from the bottom of the pot.
Add potatoes and 3 cups of the chicken broth and bring to a simmer. Cut the chicken breasts in half*, return them to the pot and simmer until the chicken is cooked through and potatoes are fork tender, 15 to 20 minutes. Remove the chicken from the pot and set it on a clean plate or cutting board to cool slightly.
While your chicken cools, in a medium pot, melt butter over medium heat. Add flour and cook, whisking continuously, until the flour is lightly golden and forms a sticky paste with the butter.
Slowly drizzle in the milk followed by the cream, whisking out any lumps as they form. Cook, whisking continuously, until mixture is thickened (it should be the consistency of thick gravy) and bubbly around the edges, about 5 minutes.
Stir the milk mixture into the soup. Shred or chop the cooked chicken and add it to the soup.
Stir in corn and peas and heat over medium heat until vegetables are heated through, 2 to 3 minutes.
If soup is too thick, add up to 1 cup more broth until it reaches your desired consistency.
Add pepper and salt to taste. Serve, and enjoy!