Set a baking sheet on the middle oven rack; preheat to 425 degrees F. Scatter the ham and cheese in the pie crust. Whisk the eggs, half-and-half, chives, 1/2 teaspoon salt and a few grinds of pepper in a large bowl; pour into the crust. Transfer to the hot baking sheet and bake until the eggs are set and the top is lightly browned, about 30 minutes.
Meanwhile, whisk the vinegar and mustard in a large bowl. Gradually whisk in the olive oil until smooth. Add the greens and mushrooms; toss well to coat. Season with salt and pepper. Cut the quiche into wedges and serve with the salad.