Make the dressing: Whisk the mustard, lemon juice and vinegar in a medium bowl. Add the capers and brine and a pinch each of salt and white pepper. Slowly whisk in the olive oil and add the tarragon. Taste for seasoning.
Bring a large pot of water to a boil, adding salt until it tastes like seawater. Meanwhile, set a colander in a large bowl of ice water (this will keep you from having to pick ice out of the peas later).
Add a generous pinch of sugar to the boiling water. Add the sugar snap peas and cook until bright green and crisp-tender, about 2 minutes. Use a strainer to remove the peas from the water and transfer them to the colander.
Bring the water back up to a boil and add the snow peas and shelled peas. Cook until they float to the surface, about 1 minute. Use the strainer to remove the peas and plunge them into the ice bath with the sugar snaps. Set aside for a couple of minutes to ensure the peas have cooled thoroughly.
Drain the peas, then spread them out on a clean kitchen towel. Use another kitchen towel to gently pat them dry, then let air-dry. (Water on the peas will dilute all of the good flavors.)
Transfer the peas to a medium bowl and season with salt, white pepper and a sprinkle of sugar. Stir to blend. Toss with the dressing and pea shoots. Taste again for seasoning and serve immediately.