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Step 1
Lightly season both sides of the chicken breast with salt and pepper.
Step 2
In a medium pot over medium-high heat, heat the olive oil until it shimmers. Lay the chicken into the pan so that it falls away from you, and cook until browned on one side, about 5 minutes. Turn over, and brown on the other side, about 5 minutes more, or until cooked through with an internal temperature of 165 degrees. Transfer the chicken to a cutting board.
Step 3
Add the onion to the pot and cook, stirring occasionally, until it starts to brown and stick to the bottom of the pot, about 5 minutes. Add a splash of the wine and stir up any brown bits. Continue cooking until the onions turn dark brown, about 4 minutes more. Add the remaining wine and, again, stir up any browned bits. Add the broth and potatoes, and bring to a boil. Lower heat to maintain a lively simmer and cook, uncovered, until the potatoes are tender, about 10 minutes.
Step 4
Meanwhile, using your fingers, pull the chicken into bite-size pieces.
Step 5
When the potatoes are cooked, remove the pot from the heat and, using an immersion blender, puree the soup on high speed until mostly smooth. (Or, using a standing blender, puree the soup in batches. Don’t overblend or it may become gluey.) Stir in the pulled chicken and cream. Taste, and season with more salt and/or pepper, bearing in mind that the cheese will also add saltiness. Just before serving, stir in the cheese.
Step 6
Ladle the soup into bowls, garnish with the chives and more pepper, if desired, and serve.