Bring the cold stock, ¼ cup salt and red potatoes to a boil in a large pot. As soon as it comes to a boil, drain immediately and spread out on a baking pan. Allow to cool completely in the refrigerator.
Heat 1 TB of oil in a 10” skillet over medium heat. Season the tuna with the house seasoning. Once oil is heated, sear the tuna for 2-3 minutes on each side, being careful not to burn. The tuna should still have a warm pink center. Do not cook past that point. If you prefer, you may cook the tuna less. Remove from the skillet and let rest at room temperature until cooled. Once cooled, shred the tuna into large pieces by hand and set aside.
In a large mixing bowl, add the shredded tuna, remaining oil and salt, onion, chilled potatoes, tomatoes, capers, olives, pepper and vinegar and combine thoroughly.
In a separate large mixing bowl, toss the mixed greens, shredded cheese and house dressing thoroughly.
Divide the salad green mixture among 6-8 plates. Divide the composed tuna salad mixture on top of each plate of greens. Garnish each with 2 quartered tomatoes, lemon wedge and additional grated cheese if desired.