To make the Bolognese sauce, heat a large pan. Add the oil with the onion, carrot, mushrooms, garlic and thyme. Cook for 8–10 minutes, until the vegetables have softened and taken on a little colour, stirring occasionally.
Add the minced beef and mix until well combined, then sauté until well browned, breaking up any lumps with a wooden spoon. Add the tomato purée and continue to cook for another minute or two, stirring.
Pour in the tomatoes and season to taste. Bring to the boil, then reduce the heat to the lowest setting and simmer for 2 hours, until the beef is meltingly tender and the sauce has reduced.
To make the cheese sauce, melt the butter in a pan. Add the flour and cook for 1 minute, stirring. Remove from the heat and gradually pour in the milk, whisking until smooth after each addition.
Season to taste. Bring the sauce to the boil, whisking constantly, then reduce the heat and simmer gently for 5 minutes, until smooth and thickened, stirring occasionally. Remove from the heat and stir in most of the Cheddar cheese until it has melted.
To cook the lasagne, preheat the oven to 180°C (350°F/gas mark 4). Brush 2 x 2.25 litre (4 pint) ovenproof dishes with the rapeseed oil, then line with a layer of lasagne sheets, breaking them to fit as necessary.
Add half of the Bolognese sauce and spread it into an even layer, then spread over half of the cheese sauce. Cover with another layer of the remaining lasagne sheets. Use the rest of the Bolognese to make another layer, then pour over the remaining cheese sauce.
Scatter the rest of the Cheddar on top and bake for 1 hour, until the lasagnes are bubbling and lightly golden. Serve straight to the table.