For the Vegetable Saute: Warm the oil in a large skillet over medium-high heat. Add the shallot and the garlic and cook until tender, about 2 minutes. Add the mushrooms and asparagus to the pan and cook until tender. Add the artichoke hearts and season with salt and pepper, then cook until the artichoke hearts are heated through. Remove the skillet from the heat, then add in the tomatoes and stir to warm them up.
For the Tarragon Vinaigrette: Combine the oil, vinegar, tarragon, salt and pepper in a glass screw-top jar. Seal the jar and shake vigorously to mix the vinaigrette.
Toss the vegetables with the vinaigrette and serve.