Preparation of Chicken: Begin by ensuring the chicken cutlets are thoroughly dried using paper towels. Season them generously with kosher salt and freshly ground black pepper on all sides.
Searing the Chicken: In a skillet of at least 10 inches in diameter, heat the olive oil over medium-high heat until it shimmers. Add the chicken cutlets in batches to prevent overcrowding, searing them until they are deeply browned on one side, approximately 6 to 7 minutes. Using tongs, turn the cutlets over and sear the other side until equally browned, an additional 6 to 7 minutes. Once browned, transfer the chicken to a separate plate and set aside.
Sauce Preparation: Lower the heat to medium and add butter to the skillet. Once the butter has melted, incorporate the minced shallot, garlic, additional salt, and pepper, cooking and stirring occasionally until the shallots begin to soften, about 1 minute. Sprinkle the flour over the shallots and garlic, stirring to coat evenly, and cook for an additional minute.
Developing the Sauce: Gradually stir in the half-and-half (or whole milk) and chicken broth, ensuring to dissolve any lumps of flour and scraping any browned bits from the bottom of the skillet. Bring the mixture to a boil, then reduce the heat to a simmer.
Final Cooking of Chicken: Return the chicken cutlets to the skillet, simmering in the sauce until it thickens sufficiently to coat the back of a spoon and the chicken is fully cooked through, about 3 to 4 minutes.
Finishing Touches: Stir in the fresh lemon juice and garnish with chopped parsley and thinly sliced lemon before serving promptly.