Let the turkey breasts stand at room temperature for 1 hour prior to cooking.
Preheat the oven to 325 degrees F. Fit a baking sheet with a wire rack.
Pat the turkey breasts dry and season heavily with salt and pepper all over. Put the olive oil in a large skillet or cast-iron pan over medium-high heat. Sear the breasts skin-side down until light golden brown, about 5 minutes. Flip the breasts over and add the garlic, thyme, rosemary and 4 tablespoons of the butter to the skillet. Once the butter has melted, use a spoon to baste the turkey breasts with the herb and garlic infused butter for 30 seconds.
Transfer the breasts to the rack on the baking sheet, pour the butter and herb mixture over the breasts and cook, basting every 15 minutes, until the internal temperature of the thickest part of the breast reaches 160 degrees F, 1 1/2 to 2 hours. Transfer to a cutting board, tent with aluminum foil and rest for 15 minutes.
Deglaze the skillet that the breasts were seared in with the white wine over medium-high heat, then reduce until almost dry, 5 to 8 minutes. Add the turkey stock and reduce by 90 percent, another 8 to 10 minutes. Reduce the heat to low and slowly whisk in the remaining 6 tablespoons butter, keeping the mixture creamy and emulsified. Once all the butter is incorporated, add the capers, parsley, lemon zest and shallot. Season with salt and pepper to taste.
Slice the turkey and transfer it to a platter. Pour some sauce over the slices and serve the rest of the sauce on the side. Garnish with fresh herb sprigs if desired.