Place the potatoes in a medium pot and cover them with cold water by about 2.5 centimeters. Bring to a boil over medium-high heat, then reduce to a steady simmer and cook for 15 to 18 minutes until very tender.
While the potatoes cook, pat the turkey cutlets dry. Season both sides with 1 teaspoon kosher salt, black pepper, dried thyme, and dried sage.
Heat a large skillet over medium heat. Add 2 tablespoons unsalted butter and the olive oil. When the butter is melted and lightly foaming, add the turkey cutlets in a single layer and cook for 3 to 4 minutes per side until golden and cooked through. Transfer the turkey to a plate and cover loosely to keep warm.
In the same skillet, add the cranberries, brown sugar, chicken broth, and lemon juice. Stir and cook over medium heat for 4 to 5 minutes, pressing some cranberries lightly as they soften, until the mixture becomes a chunky pan sauce.
Drain the potatoes well and return them to the hot pot. Add 4 tablespoons unsalted butter, warmed milk, minced garlic, and 1/2 teaspoon kosher salt. Mash until smooth and creamy, adding a small splash of extra milk if needed for your preferred texture.
Slice the turkey cutlets if desired. Spoon the mashed potatoes onto plates, top with the turkey, and finish with the warm cranberry skillet sauce.
Sprinkle chopped fresh parsley over each serving and serve immediately while hot.