Place the olives, tomatoes, garlic, artichoke hearts and 1 tablespoon of the olive oil in a food processor. Pulse until coarsely chopped, adding up to 2 more tablespoons of olive oil if needed to make a spreadable paste.
Using a sharp knife, slice each tuna steak in half horizontally to create 4 steaks about 1/2 inch thick. Heat 2 tablespoons of the olive oil in a large nonstick skillet over medium-high heat until it ripples. Season the tuna generously with salt and pepper and sear the fish on both sides until cooked through, about 2 minutes per side, adding another tablespoon of olive oil between batches if needed. Transfer the tuna to a large bowl and break into small pieces. Add the capers, onion, thyme, lemon juice and 1/4 cup of the olive oil and stir.
Spread a generous amount of the olive mixture on 5 slices of bread and top with a heaping spoonful of tuna and another piece of bread. Slice each sandwich in half diagonally.