Combine 4 tablespoons of the oil, the vinegar, tarragon, garlic and honey in a small bowl. Whisk until blended. Season with salt and pepper. The dressing can be made one day ahead. Store covered in the fridge, but serve at room temperature.
Heat the remaining 2 tablespoons of olive oil in a heavy medium skillet over medium-high heat. Add the mushrooms and sauté until wilted, about 5-7 minutes.
Place the mixed greens in a large bowl. Toss with half of the dressing. Divide the greens between the plates. Spoon the sautéed mushrooms atop the salads. Drizzle the mushrooms on each plate with a little of the remaining dressing. Serve immediately. There is something really special about hot mushrooms on top of the cool salad.