Whisk the eggs, cream, milk and parmesan and leave aside.

Heat the olive oil in a pan over a high heat, then add the mushrooms and stir until they absorb all of the oil. Cook tossing every few moments until the mushrooms become soft and turn a golden brown, then reduce the heat to medium, then add the diced onion and crushed garlic and cook until the onions start to turn translucent. 

Cook the spaghetti in a saucepan of boiling water as per the pack instructions until al dente, then drain, reserving about 125ml (1/2 cup) of the hot water.

Place the spaghetti into the pan along with the pureed carrot and reserved water and toss well until the spaghetti is fully coated.

Reduce the heat to low, then slowly pour the egg, cream and cheese into the pasta, stirring constantly until the egg turns into a glossy sauce.

Serve with a little grated parmesan for adults (omit for babies), a sprinkle of fresh thyme leaves and grated black pepper.