Prepare the meat sauce and set aside. If using previously frozen sauce, thaw about 2 1/2 cups, reheat gently, and set aside.
Make the bechamel, keeping it rather runny, about the consistency of sour cream. Keep it warm over very low heat (the upper half of a double boiler). If a film forms, stir it in before using.
Make green pasta dough (machine or hand-rolled) and roll it as thin as possible. Cut into lasagne strips as directed: machine-rolled strips cut into 10-inch lengths, or hand-rolled rectangles about 4 1/2 inches by 10 inches.
Set a bowl of cold water near the stove and lay clean, dry cloth towels flat. Bring 4 quarts (3.8 L) water to a rapid boil, add 1 tbsp salt, and when it returns to a boil slip in 4 or 5 pasta strips. Cook very briefly, just seconds after the water returns to a boil. Lift out with a colander scoop or slotted spatula and plunge into cold water. Rinse each strip under running water, rub gently, squeeze very gently, and lay flat on towels to dry. Cook the remaining pasta the same way, then pat dry on top.
Preheat the oven to 400 F (205 C).
Butter the bottom of the lasagne pan generously and spread about 1 tbsp bechamel. Cover the bottom with a single layer of pasta, cut to fit edge to edge with minimal overlap.
Mix the meat sauce and bechamel and spread a thin coating over the pasta. Sprinkle with some grated Parmigiano-Reggiano. Add another layer of pasta, then repeat sauce and cheese. Build at least 6 layers, using pasta trimmings to fill gaps if needed. Finish with a very thin layer of sauce on top, sprinkle Parmesan, and dot with butter.
Bake on the uppermost rack until a light golden crust forms, 10 to 15 minutes. If no crust begins to form after the first few minutes, raise the oven 50 to 75 degrees. Do not bake longer than 15 minutes total.
Let the lasagne settle about 10 minutes, then serve from the pan.
