To prepare fish, combine first 8 ingredients in a medium saucepan, stirring to dissolve salt and sugar. Cool to room temperature. Add Arctic char to pan. (If the fish bobs to the surface, weight it down with a small bowl.) Cover, and refrigerate 3 hours.
Soak wood chips for 1 hour. Drain.
Preheat grill, and set it up for indirect heat. For a charcoal grill, move ash-covered coals to one side of the grill, and add soaked, drained wood chips directly to the coals. For a gas grill, place the chips in a smoker box, or arrange chips on a sheet of heavy-duty aluminum foil, wrap, and poke holes in top of foil. Place the smoker box or foil packet directly on the heated burner. Heat wood chips 10 minutes, or until they start to smoke.
Remove Arctic char from brine, and discard the brine. Rinse fish thoroughly under cold running water. Pat fish dry. Brush skin side of fish with canola oil. Add fish, skin side down, to unheated side of grill. Cover, and smoke 10-15 minutes, or until desired degree of doneness. Cooking time will depend on the thickness of the filet. Flake fish into bite-size pieces.
While the fish is brining, cook the beets. Preheat oven to 400°F.
Cut tops off the beets, leaving 1 inch of the stem. Wash, pat dry and wrap each beet in foil. Bake for 30-40 minutes, or until done. (Beets will be tender when you can pierce them with a thin, sharp knife.) Let stand until cool enough to handle. Rub the skins off beets, and trim away root ends. Cut each beet into 6 wedges.
Combine salad greens, beets and vinaigrette in a large bowl, tossing to coat. Divide evenly among 4 plates. Top evenly with smoked fish and chopped egg.