Toss the endive, shallots, and radishes in a large bowl. Toss with half of the dressing and season with salt and pepper. Divide the mixture among 4 plates and arrange the beets around the plate. Tear the trout fillets into bite-sized pieces and place on each salad. Top each with 1 tablespoon of the trout roe and serve.
Whisk together lemon juice, vinegar, mayonnaise, mustards and salt and pepper in a medium bowl. Slowly whisk in the oil until emulsified and stir in the tarragon.
Combine water, salt, and sugar in a large container and mix until the salt is completely dissolved. Add the fish, cover, and refrigerate for 1 hour.
Soak wood chips in water for at least 1 hour. Remove trout, rinse and dry thoroughly.
Use a covered grill (charcoal, gas or electric) and preheat to about 200 degrees F. Cover
heated coals with 1/3 of the hickory chips. Place fish, skin side down, on well-greased grill, about 4 inches from coals. Close grill hood and open vent to circulate smoke. Add additional hickory chips, as necessary. Smoke trout for approximately 20 minutes. Trout is done when the cut surface is golden brown and flakes easily with a fork.
*You can purchase smoked trout at gourmet stores, specialty markets, and online.