Make the mayonnaise: combine all ingredients in a small bowl; mix well; cover and refrigerate at least 1 hour before using to allow flavors to blend.
Prepare a med-hot fire in a charcoal grill with a cover or preheat a gas grill to med-high.
Make the patties: combine the tuna, bell pepper, onion, Chardonnay, mustard, olive oil, and seasoned salt in a bowl.
Handling tuna as little as possible to prevent compacting it, mix well, adding just enough bread crumbs to hold the mixture together.
Divide mixture into 4 equal portions; form portions into patties to fit the buns.
When grill is ready, brush grill rack with vegetable oil; place patties on the rack; cover and cook, turning once, until done to preference, 3-4 minutes on each side for med-rare.
During the last few minutes of cooking, place the buns, cut side down, on the outer edges of the rack to toast lightly.
During last minute of cooking, place 1 slice of cheese on each patty to melt.
To assemble: spread mayonnaise over the cut sides of the buns; on each bun bottom, place an equal portion of the artichoke hearts, a few greens, and a patty; add the bun tops and serve.