Preheat oven to 450°F. Toss the rhubarb with 1 teaspoon of the oil on an aluminum foil–lined baking sheet. Bake in oven until caramelized and tender, about 4 minutes. Remove the rhubarb to a bowl and toss with the sugar. Let cool to room temperature.
Whisk together the vinegar, lemon juice, mustard, and the remaining ¼ cup of oil in a small bowl. Toss the lettuce with ¼ cup of the vinaigrette and top with goat cheese, pistachios, and rhubarb.