In a bowl, combine the vinegar, 2 tbsp and salt and pepper to taste. Add the chicken to coat. Heat a large skillet over medium heat with 1 tbs of oil. Add the chicken and cook for 5 minutes on each side. While the chicken is cooing, toss arugula with lemon, salt and pepper. Remove the chicken to a cutting board. Add the butter to the skillet and when it has melted, add the flour and cook for a minute or so. Whisk in the chicken stock and cream and bring up to a simmer to thicken. Remove from the heat and stir in the pesto. Slice the chicken and arrange on dinner plates. Pour the pesto gravy over the chicken, then top with the arugula and cheese shavings.