Cook potatoes by placing in saucepan of cold water to cover by 1 inch. Boil until just tender, about 15 minutes. Drain and cool slightly. Slice potatoes into 1/4 inch-thick rings.

Refill saucepan with water and bring to a boil. Add 1 teaspoon of salt. Boil the asparagus pieces for 2-3 minutes, until crisp-tender. Empty into colander and rinse with cold water. Drain well. Transfer to kitchen towel or paper towel to dry thoroughly.

Meanwhile, combine shallots with vinegar and allow to sit for about 5 minutes. Add mustard and basil and whisk to combine. Pour olive oil into mustard mixture in a steady stream, whisking to combine. Season with salt and pepper.

Place potatoes in large, wide bowl and add about half of the vinaigrette. Carefully use large spoon to distribute vinaigrette over surfaces of potato slices. Add asparagus, ham, and Gruyère to bowl. Pour in remaining dressing and toss to blend. Taste to see if more basil, salt, or pepper is needed.

Serve over salad greens of your choice.