Preheat oven to 200°C. Line an oven tray with baking paper.
Slice kumara into 0.5cm-thick rounds and arrange on lined oven tray. Brush or spray with olive oil. Season with salt and pepper and roast for 15 minutes or until golden (they do not have to be tender all the way through). Set aside on the bench to cool.
For the beef sauce, heat olive oil in a large fry-pan on medium-high heat. Cook beef, with a good pinch of salt, until browned, breaking up with a wooden spoon as it cooks. Set cooked mince aside. Add onion, mushrooms, eggplant and a good pinch of salt to the pan; cook for 8-10 minutes until the vegetables are soft. Stir in remaining sauce ingredients and the cooked beef mince, and simmer for 8-10 minutes until sauce is thick and most of the liquid has evaporated.
For the creamy cauliflower sauce, on a separate element heat olive oil in a medium pot over medium heat. Cook onion, with a pinch of salt, for 2-3 minutes until soft but not coloured. Add garlic and continue cooking for 1 more minute. Add cauliflower, milk and stock. Cover partially and simmer for about 15 minutes or until cauliflower is soft.
Stir in mustard and half the Parmesan, then use a blender or stick blender to puree cauliflower and its cooking liquid together until silky smooth. Season to taste with salt and pepper.
To assemble lasagne, spread ½ cup of the beef mince sauce out in the base of a medium-large baking/lasagne dish. Cover with a single layer of kumara rounds (about one-third of the kumara) then top evenly with half of the remaining beef sauce. Repeat with one more layer of kumara and beef sauce and a final layer of kumara. Finally, pour creamy cauliflower cheese sauce all over the top and sprinkle over remaining Parmesan. Bake for 25 minutes, switching to grill for the last few minutes, until golden on top.
Allow lasagne to rest for at least 10 minutes before cutting up. Serve with green salad.