In a small sauce pan over medium heat, sweat the shallots in the olive oil for 3 minutes or until soft. Add the chicken stock and the milk and bring to a gentle simmer. Sprinkle the polenta over the surface and whisk to incorporate. Simmer, stirring occasionally, for 10 minutes. Whisk in the cheeses, cream and salt and cook for 10 minutes more. Stir in the basil and parsley and keep warm.
For the pasta:
Bring a pot of salted water to a boil. Boil the lasagna sheets for 8 to 12 minutes or until al dente. Drain and lay out the pasta flat on a paper towels to dry thoroughly. Trim about 1 inch off of each end of the pasta to make all of the sheets the same size. Place the oil in a medium saucepan with a deep fry thermometer over medium high heat. Heat the oil to 375 degrees. Working in batches of two noodles at a time, fry the pasta for 1 minute per batch, flipping often, until they are golden brown and crisp all the way through. Drain on paper towel and continue with the remaining pasta.
For the vegetables:
Preheat the oven to 475 degrees.
On a sheet tray, toss the vegetables with the olive oil and salt. Place in the preheated oven and roast for 12 minutes or until cooked through and beginning to brown.
Place a fried pasta sheet on each plate. Top with some of the roasted vegetables and spoon about 1/4 cup of the béchamel over them. Repeat with another pasta sheet, vegetables and sauce. Finish with a fried pasta sheet, the remaining sauce and some freshly grated Parmesan.