Trim any excess fat from the 2 boneless, skinless chicken breasts and slice each breast in half lengthwise to make four thin cutlets. Pat dry with paper towels.
Combine the 3 tablespoons all-purpose flour as follows: use 2 tablespoons to dredge the chicken (place flour on a plate and coat both sides of each cutlet, shaking off excess) and reserve the remaining 1 tablespoon for the sauce.
Season both sides of the floured chicken cutlets evenly with the ½ teaspoon kosher salt and ¼ teaspoon freshly ground black pepper (reserve additional salt/pepper to adjust seasoning at the end).
Heat a large skillet over medium-high heat. Add 1 tablespoon extra virgin olive oil and 1 tablespoon of the butter. When the fat is shimmering, add the chicken cutlets in a single layer (work in batches if needed) and cook 3–4 minutes per side, until golden brown and cooked through. Transfer the cooked cutlets to a platter and cover loosely with foil to keep warm.
Reduce the heat to medium. Add the 2 cloves garlic (pressed or minced) to the skillet and cook, stirring, until fragrant, about 30–60 seconds. Sprinkle the reserved 1 tablespoon flour over the garlic and stir continuously for 1 minute.
Whisk in 1 cup chicken stock, the juice of 1 lemon, and 4 tablespoons heavy cream (or half-and-half), scraping up any browned bits from the bottom of the pan. Bring the sauce to a gentle simmer.
Add the ½ lemon thinly sliced to the sauce and simmer 4–5 minutes, stirring occasionally, until the sauce thickens slightly. Stir in the remaining 1 tablespoon butter until melted and incorporated.
Return the chicken cutlets to the skillet, spoon the sauce over them, and simmer 1–2 minutes to heat through and meld flavors.
Stir in 1 tablespoon chopped fresh thyme leaves, taste the sauce, and adjust seasoning with additional kosher salt and freshly ground black pepper as desired. Serve the chicken with the lemon slices and sauce spooned over the top.