Remove the tips and the first two joints of the wing bone from the chicken breasts, leaving the bone attached.

Place chicken breasts between two pieces of plastic wrap.

Pound the breasts to 1/2-inch thickness, being careful not to separate the meat from the bone.

Pat the chicken breasts dry and season with salt and pepper.

Dredge the chicken breasts in seasoned flour, shaking off the excess.

Dip the chicken breasts in the beaten egg, letting the excess drip off.

Coat the chicken breasts with the bread crumbs and Parmesan cheese mixture.

In a large heavy skillet, heat vegetable oil and butter over moderately high heat.

Add the quartered garlic cloves to the skillet and sauté until golden; then discard the garlic.

Sauté the chicken breasts in the flavored oil and butter, turning once, for about 4 minutes per side, or until golden brown and cooked through.

Transfer the cooked chicken breasts to paper towels to drain excess oil.

To make the salad dressing, whisk together white wine vinegar, Dijon mustard, dried thyme, salt, and pepper in a small bowl.

Slowly drizzle olive oil into the bowl while whisking continuously until the dressing is emulsified.

In a separate bowl, toss the arugula, cherry tomato quarters, and grated carrot with a portion of the salad dressing.

Arrange each chicken breast on a plate.

Divide the arugula salad between the plates.

Garnish each serving with a lemon wedge.