Fill a large bowl with cold water. Cut 2 of the lemons in half and juice them into the water, tossing in the lemon rinds as well. Remove the tough outer leaves from each of the artichokes so only the tender, light yellow leaves are showing. Use a paring knife to trim the bottom and to peel off the green outer layer from the stem. Trim the top 1 inch from each artichoke. Put each artichoke in the acidulated water as soon as you are finished trimming it.
When all of the artichokes have been trimmed, cut them in half lengthwise and use a spoon to scoop out the furry choke and the tiniest pinkish leaves. Place the cleaned artichokes in a medium saucepan and cover with cold water. Add the vinegar, bay leaves, and about 1½ tablespoons salt; the water should be just slightly less salty than the sea. Place a piece of parchment paper cut to fit just inside the pan over the artichokes, pressing it down so it touches the water. Bring to a boil over medium-high heat. Reduce the heat and simmer until the artichokes can be easily pierced through the heart with a knife, about 30 minutes. Remove the pot from the heat, remove the parchment from the pan, and let cool completely.
Bring a pot of salted water to a boil and prepare a bowl of ice water. Snap the fibrous ends off the asparagus. Add the asparagus to the boiling water and cook until slightly tender but still with good bite, about 1 minute. Drain the asparagus and plunge it into the ice water to stop the cooking and retain the green color. When cool, dry the asparagus on a clean kitchen towel. Cut on a slant into 1½-inch pieces. Set aside.
Juice the remaining lemon and mix with the shallot in a small bowl. Set aside to macerate for 10 minutes. Add a pinch of salt and, whisking constantly, slowly pour in the olive oil. Set the dressing aside.
Remove the artichokes from their cooking liquid and slice them thinly lengthwise from top to stem. Combine the sliced artichokes and the asparagus in a medium bowl and set aside.
Prepare the grill so the coals are glowing red but no flame is visible. Trim any excess fat from the steaks and season them with salt and pepper. Grill for 2 to 4 minutes on each side, depending on their thickness, for medium rare. Set the steaks aside on a warm plate to rest.
Strip the leaves of mint from the stems and coarsely chop the leaves. Put them in the bowl with the artichokes and asparagus. Add the dressing and toss lightly to combine. Season to taste with salt and pepper.
Arrange the salad on 4 plates. Slice the steaks against the grain and on a slant. Arrange the slices of meat on the plates with the salad, pouring any juices on top. Drizzle with olive oil and serve.