Put the vinegar, mustard, egg yolk, garlic, 1 teaspoon salt, and 1/2 teaspoon pepper in a small bowl or 2-cup glass measuring cup. Slowly whisk in the olive oil, making an emulsion. Place the arugula in a large salad bowl, pour on enough vinaigrette to moisten, and toss well.
Distribute the arugula on 4 dinner plates. Place half a burrata, cut side down, in the middle of the plate and artfully arrange the prosciutto around it (I tear each slice of prosciutto in half lengthwise). Sprinkle with salt and pepper, and serve.