To make the vinaigrette, on the stove top or grill, melt the butter in a saucepan over medium heat, then turn the heat to medium-low and simmer gently, swirling the pan often, until the butter is browned and smells nutty, about 10 minutes. Let cool to room temperature. In a large bowl, whisk together the brown butter, vinegar, shallot, garlic, and Dijon. Whisk in the oil and season to taste with salt and pepper. (You can also puree it in a blender if you prefer.)