Finely chop sun-dried tomatoes in a processor. Blend in mayo and mustard. Transfer to small bowl. (Can be made hours ahead. Cover; refrigerate.)
Finely chop andouille sausages in processor. Transfer to large bowl. Add turkey, shallots, salt, pepper, and crushed fennel seeds. Stir with fork just until blended. Form mixture into six 1-inch-thick patties.
Prepare barbecue (medium-high heat). Grill hamburger buns until golden, about 2 minutes. Transfer to platter. Brush onion slices with oil. Sprinkle with salt and pepper. Grill until golden, about 7 minutes per side. Grill hamburgers to desired doneness, about 5 minutes per side for medium-rare. Put Swiss cheese over top of burgers.
Spread cut sides of hamburger buns with sun-dried-tomato mayonnaise. Top bottom halves of buns with hamburgers, then red peppers. Top with onion slices. Cover with top halves of buns and serve with your favorite chilled tea.