Make the salad: Preheat oven to 350 degrees. Wrap red beets in parchment, then in foil. Repeat with yellow beets. Bake until tender, 60 to 65 minutes. Let stand until cool enough to handle. Peel, and halve (or quarter larger beets). Place red and yellow beets in separate bowls.
Add peas to bowl with yellow beets. Sprinkle red and yellow beets with the thyme, salt, and pepper.
Make the dressing: Heat 1 teaspoon oil in a medium skillet over medium heat. Add onion, and cook, stirring frequently, until caramelized, about 30 minutes. Stir in thyme and 1 tablespoon vinegar. Cook for 2 minutes. Transfer to a food processor, and puree. Add water, salt, pepper, and remaining 2 teaspoons oil and 1 tablespoon vinegar. Puree until smooth. (Dressing should be thick.)
Toss 1/2 of the dressing with the red beets and 1/2 with the yellow beet mixture. Combine beets, and gently toss with shoots. Garnish with pea flowers if desired, and serve immediately.
Sugar snap peas, a cross between English peas and snow peas, are sweetest immediately after harvest. Pea shoots are baby pea plant tendrils. The freshest samples are found at farmers' markets.