Preheat oven to 400 degrees.
Place potatoes in a medium sauce pan, cover with cold water and add a pinch of salt. Bring to a boil, cover and simmer for 20 minutes or until potatoes are tender.
Slice chicken breasts open like a book, being careful not to cut all the way through. Cover with plastic wrap and point to an even thickness using a meat mallet or heavy skillet.
Tear each slice of cheese in half and lay one half on both sides of each chicken breast. Divide sundried tomatoes and basil between one half of each chicken breast. Carefully fold the chicken breasts closed. Season outside with salt and pepper.
Heat oil in a 10 inch skillet. Add chicken breasts and sear for 2 to 3 minutes. Carefully flip and then transfer the pan to the oven for 10 to 12 minutes until the chicken reaches 165 degrees.
In a small, 8 inch, skillet start the sauce by heating some olive oil over medium low heat. Add garlic and cook for 30 seconds or until fragrant. Melt the butter and then add the wine. Bring to a simmer and reduce by about half.
Drain potatoes. Add butter, oil from sundried tomatoes, and buttermilk. Mash potatoes roughly. Add provolone cheese and mash to desired consistency. Stir in sundried tomatoes.
To serve, set chicken on top of mashed potatoes and spoon over some of the white wine and butter sauce.