Bring a large pot of salted water to a boil. Cook the potatoes then the beans in the water, cooking each until just tender. Drain the potatoes and slip off the skins. Drain the beans and refresh them in ice water.
Refill the pot with salted water. Cook the pasta sheets 1 at a time, just until tender. Drain each sheet and lay on clean towels. Cover with additional towels to keep warm.
Coat the bottom of a 9 by 13-inch serving dish with a thin layer of pesto. Cover the pesto with a 1/3 of the warm lasagne noodles. Spread pesto over the pasta. Break the potatoes up into bite-size pieces and put half of them and half the beans on top. Arrange another layer of pasta over the beans and potaotes, cover with pesto, another layer of beans and potatoes then a final layer of pasta. Cover the lasagne with Parmigiano-Reggiano, cut and serve.
Place the basil leaves, garlic, and pine nuts in a mortar or food processor. Grind or pulse until the leaves are finely chopped. Gradually incorporate the Parmigiano Reggiano, ricotta and yoghurt. Then, work in the oil, adding it in a steady stream. Season the pesto to taste with salt and pepper.
Mix the flour with the salt and mound them on a clean work surface. Create a well in the center of the flour mound. Add the eggs and 1 tablespoon of olive oil to the well. Lightly beat the eggs with the oil. Using the fingers of one hand, begin incorporating the dry ingredients into the wet, drawing flour into the well in a circular motion. Use the other hand to protect the outer wall. Work the flour mixture into the egg mixture until the dough forms a ball (if the dough doesn't come together add a drop or two of water). Knead the dough until it is smooth and elastic, about 10 minutes. Brush the surface with the remaining olive oil , wrap the dough in plastic wrap; and set aside to rest for about 30 minutes.
Cut the ball of dough in half. Wrap and reserve the piece you are not immediately using to prevent it from drying out. Dust the counter and dough with a little flour. Form the dough into a rectangle and roll it though a pasta machine, 2 or 3 times, at the widest setting. Pull and stretch the sheet of dough with the palm of your hand as it emerges from the rollers. Reduce the setting and crank the dough through again, 2 or 3 times. Continue tightening until the machine is at the narrowest setting: the dough should be paper-thin, about 1/8-inch thick. Make 4 pieces, about 20-inches long and 6-inches wide. Lay the lasagne sheets out on a clean dry surface and allow them to dry slightly (about 10 minutes) before using.