Heat 2 tablespoons olive oil in a medium skillet over medium-high heat. Add shallots and sauté for 4 minutes. Add apples and sauté for an additional 4 minutes, stirring frequently. Add chestnuts and sauté for another 2 minutes. Remove from heat and add the sherry vinegar to the pan, and then the remaining olive oil along with a pinch of salt and pepper. Stir to combine.
Push washed greens in a bowl and spoon chestnut mixture atop greens, and crumbled blue cheese. Toss to combine. Sprinkle with parsley to garnish and serve.