Marinate the chicken in 1/4 cup of olive oil, crushed garlic cloves and thyme; refrigerate.
Cut the onion in half, then thinly slice into half-moons and place the onions in a bowl of ice water with a sprinkle of salt; set aside.
In a bowl, whisk together the red wine vinegar, Dijon mustard and 1/3 cup olive oil until emulsified; reserve.
Clean the mushrooms and slice into bite-size pieces. Heat a large pan over high heat, when hot, add the remaining 2 tablespoons of extra virgin olive oil, followed immediately by the mushrooms. Arrange the mushrooms in a single layer and allow to sear, undisturbed. Once the mushrooms have nicely browned, season with salt and deglaze with a splash of brandy. Cook until the pan is dry and remove the mushrooms to a plate; reserve.
When ready to prepare the chicken, shake off any excess oil, season with salt and grill on a hot grill pan until nicely marked. Flip the chicken and grill the other side; reserve.
Drain the onions through a colander and rinse under cold running water.
Arrange the lettuces in a bowl, add the mushrooms and the onions and season with salt. Dress the greens with the vinaigrette and taste for seasoning. Arrange the salad among the plates, top with the grilled chicken, and finish with some shaved Parmigiano-Reggiano cheese. Serve immediately.
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