Fry the beef in 4 teaspoons of oil until brown. Add the onions and garlic and sauté. Add the tomato paste and fry until the mixture is almost dry. Add the tomatoes, stock, and herbs, season, and cook until nearly dry.
Bring a pan of salted water to a boil, add the remaining oil, and cook the lasagne for 5 minutes, then rinse in cold water.
Cook the beans in boiling salted water until tender, and rinse. Melt 4 teaspoons butter, stir in the flour and cook, stirring for 1 minute. Stir in the milk and simmer, stirring, for 15 minutes. Season and strain.
Grease an ovenproof dish. Cover with a layer of lasagne sheets, a layer of beef, beans, sauce, and cheese. Layer until the ingredients are used, ending with pasta topped with the sauce and sprinkled with cheese.
Bake in a preheated oven at 350°F for 40 - 45 minutes. Cut into squares to serve.