Season both sides of the turkey cutlets with kosher salt and freshly ground black pepper.
Heat olive oil in a large skillet over medium-high heat until shimmering.
Add the turkey cutlets to the skillet and cook for 3 to 4 minutes on each side, or until golden brown and cooked through. Remove the cutlets from the skillet and set aside on a plate.
Reduce the heat to medium and add 2 tablespoons of butter to the skillet.
Add the minced garlic and chopped shallot to the skillet and sauté for 1 to 2 minutes, stirring frequently, until fragrant and softened.
Pour in the dry white wine and scrape the bottom of the skillet with a wooden spoon to loosen any browned bits. Simmer for 3 to 4 minutes until the wine is reduced by half.
Add the chicken broth and continue to simmer for another 4 to 5 minutes until the sauce has slightly thickened.
Stir in the heavy cream and fresh thyme leaves, and cook for an additional 2 minutes, stirring occasionally.
Return the turkey cutlets to the skillet and spoon the sauce over them. Cook for 1 to 2 minutes to warm the cutlets through.
Remove the skillet from heat and stir in the remaining 1 tablespoon of butter until melted and the sauce is glossy.
Garnish with chopped fresh parsley before serving.