Tip:To quickly toast several slices of bread at once, use your broiler. Set the rack as close to the element as possible, arrange the bread in a single layer on a heavy-duty baking sheet, and toast both sides under the broiler. Watch carefully to avoid burning.
Cut the tuna into even 3- to 4-inch pieces. In a deep saucepan, wide enough to hold a few tuna pieces without crowding and deep enough to contain the poaching oil without spilling when the tuna is submerged, heat the olive oil over medium heat until an instant-read thermometer inserted into the oil reaches 190°F. The oil should stay at or near this temperature during the cooking process, so you'll need to monitor and adjust the intensity of the heat under the pan.
Rub the tuna with 1 Tbs. of the lemon juice and season with salt and pepper. Carefully submerge as many pieces of the tuna as will fit in a single layer in the hot oil. The oil must cover the tuna, so add more oil if needed. Cook until the tuna turns pinkish gray on the outside and is slightly pink in the center (remove a piece of tuna with a slotted spatula or spoon and cut into it to check), 3 to 6 min. Remove the tuna from the oil and set on a platter to cool. Repeat the process as necessary to cook the remaining tuna. Don't discard the poaching oil yet.
When the tuna is cool enough to handle, break it into small chunks over a medium bowl. Add the artichokes, capers, 1/4 cup of the poaching oil, and the remaining 3 Tbs. lemon juice. (Discard the remaining poaching oil.) Season with 1 tsp. salt and several grinds of pepper or to taste. Toss gently and let the tuna salad sit for 10 min. at room temperature or overnight in the refrigerator. Before serving, bring to room temperature and check the seasonings.
Right before you leave for the picnic, or when you reach your picnic destination, layer a lettuce leaf, a tomato slice, and about a sixth of the tuna salad between two slices of toast (or use less if a sixth of the salad seems like too much for the size of your bread slices). Repeat to make six sandwiches. Wrap well in plastic for transport or serve immediately.