Whisk lemon juice and mustard in a large bowl. Gradually add oil, whisking constantly until emulsified; season vinaigrette with salt and pepper.
Working in batches, cook green peas and sugar snap peas in a large pot of boiling salted water until crisp-tender, about 2 minutes per batch. Immediately transfer to a bowl of ice water and swoosh peas around until cold; this sets their color and halts the cooking. Drain and pat dry with paper towels.
Add green peas, sugar snap peas, and arugula to bowl with vinaigrette and toss until well coated with dressing; season with salt and pepper.
Arrange salad on a platter and top with prosciutto. Finely grate horseradish over salad to your liking.