Sautéed Chicken Paillards with Herb Salad and White Balsamic Vinaigrette

1 Tbs. Celia's Traditional white balsamic vinegar
1-1/2 tsp. white wine
One-half small shallot, finely chopped
1/4 tsp. Dijon mustard
Kosher salt and freshly ground black pepper
3 Tbs. plus 2 tsp. olive oil; more as needed
8 boneless, skinless, thin-cut (1/4- to 1/2-inch-thick) chicken breast cutlets (1-1/2 to 1-3/4 lb.)
5 cups mixed baby greens
3 cups mixed fresh, tender herb leaves, such as flat-leaf parsley, mint, chives, tarragon, basil, and chervil, roughly torn if large

In a small bowl, mix vinegar and wine with the shallot, mustard, and 1/4 tsp. each salt and pepper. Slowly whisk in 3 Tbs. of the oil.
Season the chicken on both sides with 1 tsp. salt and 1/2 tsp. pepper. Heat the remaining 2 tsp. oil in a 12-inch skillet over medium-high heat until hot. Working in two to three batches to avoid crowding, cook the chicken until lightly browned on both sides and just cooked through, 1 to 3 minutes per side. Transfer each batch to a platter, cover loosely with foil, and keep warm. If the pan seems dry at any point, add 2 more tsp. of oil.
Combine the greens and herbs in a large bowl. Add about three-quarters of the vinaigrette and toss well. To serve, arrange two cutlets on each serving plate and drizzle the remaining dressing over the chicken. Divide the herb salad among the four plates, piling it attractively onto the chicken.