Place the garlic, thyme, 1 teaspoon salt, and 1/2 teaspoon pepper in a food processor fitted with the steel blade and process until the garlic is finely minced. Add the panko, lemon zest, 1 tablespoon olive oil, and the butter and pulse a few times to moisten the panko. Pour the mixture into a shallow bowl. In another shallow bowl, whisk together the mustard and wine.
Pat the chicken thighs dry with paper towels and sprinkle generously with salt and pepper. Dip each piece first in the mustard mixture to coat the top and bottom and then coat the skin side only in the crumb mixture, pressing gently to make the crumbs adhere. Place the chicken on one side of a sheet pan, crumb side up, and press any remaining crumbs onto the chicken.
Place the potatoes, 1 tablespoon olive oil, 1 teaspoon salt, and 1/2 teaspoon pepper in a bowl and toss. Spread the potatoes on the other side of the sheet pan in one layer and roast both together for 40 to 50 minutes, turning the potatoes once during roasting, until the chicken reaches an internal temperature of 160 degrees F. The chicken and potatoes should be done at the same time.
Meanwhile, place the greens on a serving platter. When the chicken and potatoes are done, toss the salad with enough vinaigrette to moisten and place the chicken and potatoes on top, adding any crumbs from the sheet pan. Sprinkle with salt and serve while the chicken and potatoes are still warm.
Place the shallots, vinegar, olive oil, mustard, 1 teaspoon salt, and 1/2 teaspoon pepper in a small bowl and whisk until emulsified.