In a large pot, bring 3 quarts of water and 1/4 cup of the salt to a boil.
Place the artichokes on a cutting board. With a sharp chef's knife, trim the stem ends off until the bottom is flat. This way, the artichokes can sit upright. Cut the top third from each artichoke and discard.
With kitchen scissors, cut the pointed tip from each of the remaining outer leaves. Discard.
Rub the cut areas with the lemon halves.
Squeeze any remaining juice from the lemon halves and add the juice to the boiling water along with the juiced lemon halves themselves.
Using a slotted spoon or skimmer, carefully add the artichokes to the boiling water. Top with a heavy pot lid or heat-proof baking dish that will fit inside the pot, so that the weight will keep the artichokes submerged in the water. This is a bit tricky. Be sure to have an adult help with this.
Lower the heat and cook at a slow boil until the artichokes are tender, about 20 to 30 minutes. With oven mitts or pot holders, carefully remove the lid or weighted dish from the top of the artichokes.
With a slotted spoon or skimmer, and with a small bowl or dish held underneath to catch the hot water, carefully transfer the artichokes to a colander set in a sink. With tongs, turn each artichoke upside down and place in the colander. Let drain, inverted, until cool enough to handle.
To make the lemon-butter sauce, melt the butter over medium heat in a small saucepan.
Add the remaining 2 tablespoons of lemon juice and the remaining teaspoon of salt, and stir to combine.
Remove the lemon-butter sauce from the heat and transfer to small dipping bowls to serve with the artichokes.