Preheat oven to 400 degrees. Arrange prosciutto in a single layer on a rimmed baking sheet and brush with oil. Bake until crisp and deep golden, about 9 minutes. Let cool on a paper-towel-lined plate.
Meanwhile, bring a medium pot of generously salted water to a boil. Add asparagus and cook until just tender, 1 to 2 minutes. Transfer to a bowl of ice water to stop the cooking; drain, then cut into 2-inch pieces on the bias. Add peas to water and cook until just tender and bright green, 3 to 4 minutes. Transfer to ice water to stop the cooking; drain. Add potatoes to water and cook until just tender, 6 to 8 minutes. Drain and let cool slightly.
Whisk together buttermilk, mayonnaise, shallot, and lemon juice in a bowl. Season with salt and pepper. Combine asparagus, peas, potatoes, tarragon, and spinach in a large bowl. Season with salt and pepper. Divide among 4 plates and top with crisped prosciutto. Drizzle with 2 tablespoons dressing and serve with remaining dressing on the side.