Light the grill. In a small bowl, combine the 3 tablespoons oil and the thyme. Coat the chicken with about 1 tablespoon of the thyme oil and sprinkle with 1/4 teaspoon of the salt and 1/8 teaspoon of the pepper. Grill the chicken over moderately high heat until just done, about 4 minutes per side.
Remove and let rest for 5 minutes, and cut diagonally into 1/4-inch pieces.
In a medium bowl, toss the mushrooms and carrots with the remaining thyme oil, 1/4 teaspoon of the salt, and 1/8 teaspoon of the pepper. Grill the vegetables over moderately high heat, turning, until just done, about 4 minutes per side for the carrots and 6 minutes per side for the mushrooms.
In a small glass or stainless-steel bowl, whisk together the mustard, lemon juice, and the remaining 1/2 teaspoon salt and 1/2 teaspoon pepper.
Whisk in the remaining 1/3 cup oil.
In a large bowl, combine the lettuce, half of the scallions, and all but 2 tablespoons of the vinaigrette. Mound onto plates. Top with the vegetables and chicken.
Drizzle the remaining vinaigrette over the chicken and top with the remaining scallions.
Wine Recommendation: Look for a wine that has plenty of acidity to stand up to the vinaigrette. In the warmer months, a white such as a California sauvignon blanc or an Italian pinot grigio will taste best. If you prefer a red wine, try a gamay or pinot noir from California.
