With a sharp knife, make a vertical cut down the center of each chicken breast, stopping just short of cutting through. Gently open one side, slicing with short, controlled cuts until it can lie flat. Repeat on the other side to form a thin, even paillard.
Place the chicken between two sheets of parchment or plastic wrap. Gently pound with the flat side of a meat mallet or heavy pan until evenly thin. Repeat with the second breast.
Season both sides of each paillard with salt and pepper.
In a large skillet, heat olive oil and butter over medium-high heat, until the butter begins to foam. Add the chicken and sauté for about 2-3 minutes per side, or until golden brown and cooked through. Transfer to serving plates.
In a medium bowl, whisk together the Greek yogurt, Dijon mustard, olive oil, lemon juice, anchovy paste, and grated garlic. Gradually add water until the dressing reaches your desired consistency. Season with salt and pepper to taste.
Add romaine lettuce to a large bowl. Drizzle with the dressing and toss until evenly coated.
Plate the chicken alongside the Caesar salad. Garnish with freshly grated Parmesan and serve immediately. Store extra dressing in the fridge for 1 week.
