Whisk the eggs and olive oil together in a small jug. Sift the flour onto a clean work surface, sprinkle over the salt and make a well in the centre.
Pour about half of the egg mixture into the well and, using your fingers, gradually draw the flour into the egg. Mix evenly, adding more egg as you go until you can bring it together to form a dough.
Knead for 10 minutes until the dough is smooth and elastic. Shape into a ball, wrap in cling film and chill for 30 minutes.
Divide the chilled dough into 4-5 equal pieces. Put 1 piece onto a lightly floured surface, and cover the rest with a clean, damp cloth. To roll out by hand, roll using a rolling pin until as thin as possible. If using a pasta machine, shape the dough into a rectangle, flattening it so it fits through the pasta machine. Flour the rollers, then feed the dough through on the widest setting and onto the floured surface. Continue to feed through the machine, over and over again, reducing the width of the roller until you get a thickness of about 2mm.
Each lasagne sheet should be about 15cm in length. Cover with another clean damp cloth and repeat with the remaining dough to make 4-5 sheets.
Beef Filling and Cheese Sauce
Preheat oven to 180°C.
Brush a shallow ovenproof dish with melted butter or oil, line with lasagna sheets – breaking them to fill any gaps. Set aside.
To make meat sauce, heat oil in a large pan. Add onion and garlic and stir over low heat until onion is tender. Add mince. Brown well – breaking up with a fork as it cooks. Stir in crushed tomatoes, lemon juice, basil, salt and pepper (to taste). Bring to boil. Reduce heat, simmer 20 minutes.
To make cheese sauce, melt butter in a medium pan. Add flour and sit for one minute. Remove from heat. Gradually add milk, stirring until mixture is smooth. Return to heat. Cook, stirring constantly, over medium heat until sauce boils and thickens. Reduce heat, simmer for three minutes. Remove from heat, add cheese, salt and pepper. Stir well until combined and set aside.
Spoon one-third of the meat sauce over lasagna sheets. Top with one-third of the cheese sauce. Arrange a layer of lasagna sheets on top.
Continue layering, finishing with lasagna sheets. Sprinkle with cheese.