Cut fingerling potatoes on the bias into 1/4-inch-thick slices. Blanch in boiling water for 4-5 minutes, until just tender but still holding their shape.
In a skillet, heat olive oil over medium-low heat. Add shallot, blanched potatoes, salt and pepper. Cook until the potatoes have started to brown and are soft, 10 to 12 minutes. Stir in rosemary; cook for a minute more.
Combine the arugula, potatoes, walnuts and Parmesan. In a small bowl, whisk together dressing ingredients; pour over the salad, and toss until everything is well-combined.
Notes
*If you can find parm made with vegetarian rennet, sub in your favorite veg-friendly cheese.