Preheat the oven to 450 degrees F.
Quarter and core the apples. Dice half of them and set aside. Thinly slice the remainder lengthwise and set aside.
In a medium sautee pan, heat the olive oil over medium-high heat. Add the carrot, celery, shallots, and the diced apples and sautee until softened, about 5 minutes. Transfer to a bowl and add the crumbled corn bread, then add the chicken broth, melted butter, thyme, and chopped pecans and toss thoroughly with your hands to combine. Season with salt and pepper.
Sprinkle the cavity of each quail with salt and pepper, then stuff a little of the corn bread mixture inside. Season the outside of the quail with salt and pepper and tie the legs together with kitchen twine.
Heat the olive oil over high heat in a heavy ovenproof sautee pan just large enough to hold the quail without touching each other. Add the quail and sear, turning occasionally, until golden brown, 4 to 6 minutes. Transfer the pan to the oven and roast the quail, for 6 to 9 minutes; the breast meat should still be rosy in color.
Meanwhile, in a medium bowl, toss the lettuce with the remaining apples and pecans and just enough of the vinaigrette to coat.
Remove the string from the quail. Arrange the salad on serving plates and place the quail alongside. Drizzle a little more vinaigrette over each quail and serve.