Roast the Tomatoes: Preheat your oven to 375°F. On a small rimmed baking sheet, toss the halved cherry tomatoes with 3 tablespoons of olive oil, the minced garlic, and a pinch of salt. Roast for 15 minutes until slightly soft and blistered, then set aside.
Dredge the Chicken: In a shallow dish, whisk together the ½ cup of flour, ¼ teaspoon salt, and ¼teaspoon black pepper. Season the chicken cutlets directly with the remaining salt and pepper, then lightly coat each piece in the flour mixture, shaking off any excess.
Sauté the Cutlets: Heat the remaining 3 tablespoons of olive oil in a large skillet over medium-high heat. Working in batches to avoid overcrowding, cook the chicken for 2 to 3 minutes per side until golden brown and cooked through. Transfer to a warm platter.
Deglaze the Pan: Pour the white wine into the hot skillet, then use a wooden spoon to scrape up any browned bits stuck to the bottom. Bring to a rapid boil and let reduce by half.
Simmer the Sauce: Stir in the chicken stock, fresh lemon juice, and drained capers. Return the chicken cutlets to the skillet, reduce the heat to low, and let them simmer gently for 3 to 4 minutes to absorb the flavors. Transfer the chicken back to your serving platter.
Finish and Serve: Dust the cold cubed butter with 1 teaspoon of flour, then whisk the butter and chopped parsley directly into the simmering pan sauce until melted and velvety. Gently stir in the quartered artichoke hearts and roasted cherry tomatoes just long enough to warm them through. Spoon the sauce, artichokes, and tomatoes generously over the chicken, garnish with lemon slices, and serve immediately.