Heat the oil in a frying pan, and add the chopped leek, diced mushroom, grated carrot, and minced garlic. Cook over a medium heat, stirring regularly, for 5 minutes.
Meanwhile, melt the butter in a small saucepan over a fairly low heat. Add the flour, and whisk well for a minute or two. Next, add the beer a little at a time, whisking until smooth each time before adding more. Add the milk in the same way, and season generously. When the sauce has come together, add the mustard and grated cheese, and cook for a couple more minutes until the cheese has melted. The sauce should be smooth and fairly thick (see photo above).
Heat the oven to 190°C (Gas Mark 5 / 375°F).
In a medium-sized baking dish, add a tablespoon or so of tomato sauce, and spread it around - just enough to lightly coat the bottom of the dish. Layer up the lasagne sheets, tomato sauce, vegetables, and rarebit sauce in alternating layers, until all the ingredients have been used - the exact order doesn't really matter. I finished mine with the last few vegetables, and the last few dollops of rarebit sauce. Spread the sauce around gently with the back of a spoon until it nearly covers the baking dish.
Bake the lasagne for around 30 minutes, until the sauce on top is golden brown, and the pasta is tender (test it with a sharp knife). If the topping is browning more quickly than the pasta is softening, just cover the dish with foil and continue cooking.
When the lasagne is cooked, allow it to stand for at least 5 minutes before cutting it into portions and serving, topped with fresh parsley if desired.