Preheat the oven to 400 degrees.
Prepare a bowl of acidulated water by squeezing the juice of one lemon to the water.
To prepare the artichokes, snap off the outer leaves down to the tender inner leaves. Slice off the tops and quarter the artichokes, rubbing the cut edges with half a lemon to prevent them from discoloring. Cut out the fuzzy inner choke and thinly slice each quarter. Drop into the bowl of acidulated water. Repeat with all the artichokes.
Drain the artichokes and put them into a large flat pan. Add the butter and water just to cover. Cook over a medium-high fire until the water has totally evaporated and the artichokes are tender. Salt well and transfer to a large bowl. Mix with 3/4 quarters of the Béchamel. Taste, add the juice of the remaining half lemon, and more salt and pepper if necessary.
Htydrate the sun dried tomatoes in hot water.
Put the cheeses in two separate bowls. Fill another large bowl with cold water. Fill yet another pan with water, add some salt, and bring to a boil. Butter a baking pan large enough to hold the finished dish.
To assemble, butter a large baking pan. Drop enough pasta to cover the bottom of the baking dish into the boiling water. After a minute, remove the pasta with a slotted spoon and drop into a bowl of cold water to cool. When the pasta is cool, remove and layer the bottom of the baking dish with the noodles. Spread 1/4 of the artichoke cream over, then 1/4 of each of the cheeses and some tomato pieces.
Repeat the process to make two more layers. Put another layer of pasta leaves on top, and cover with the remaining Béchamel and cheeses. Add some more salt, and grind some pepper over the top. (Up to this point, the lasagna can be prepared some hours before serving.).
Bake for about half an hour, until a golden-brown crust has formed. Allow to rest a bit before serving.