Preheat oven to 400°F.
Toss the quartered beets with olive oil. Wrap them in foil and roast for 20 minutes or until tender. Remove from oven, open foil, and let beets cool. When they are cool enough to handle, peel off the skins.
While the beets are roasting, place the sugar snap peas and haricot verts in a heat-safe bowl large enough to hold the peas plus about 2 to 3 times the amount of water.
Bring a large pot of water to a rolling boil, drop in the frozen edamame. After the pot comes back up to a boil, count to 60, then drain the edamame into a colander/strainer over the bowl of sugar snap peas and haricot verts.
Set the edamame aside to cool. If not already, shell them.
Drain the sugar snap peas and green beans and allow to cool.
For each serving, place a handful of arugula on a salad plate. To each, add four beet sections, edamame and sugar snap peas and green beans. Drizzle with dressing.