Place the beetroot in a microwavable bowl and add 30ml of water. Cover with clingfilm, then place in a microwave. Cook on high for 10 minutes until the beetroot is just tender. Leave to cool completely, then strain the juice, reserving it for later.
Wash and thoroughly rinse the sushi rice. Add to a saucepan with 330ml water and bring to the boil. Reduce the heat and simmer for 10 minutes with the lid on, then turn off the heat and leave to stand for 25 minutes, without removing the lid. Allow the rice to cool slightly, then add the rice vinegar and caster sugar and season, folding through. Add 2-3 tbsp beetroot juice, depending on how pink you want the rice to be. Fold through until evenly coloured, then set aside.
Whisk together the eggs, milk, soy sauce and sugar. Add the sunflower oil to a 20cm non-stick frying pan and place over a medium/low heat. Use a pastry brush to brush the oil all over. Reduce the heat to low and add half the egg mixture, swirling around so it coats the pan. When the omelette is wet but the edges and bottom have dried, fold in the right and left side by about 3cm, and then fold in the top twice, leaving the bottom half of the omelette unfolded. Use a spatula to gently nudge the omelette so that the twice-folded side is resting against the side of the pan.
Add the remaining egg mixture around the omelette. Allow to set slightly, then fold in the sides as before and continue to fold in the top edge as it sets, until you have a rectangle of omelette. Flip over and press on each side, then transfer to a plate and leave to cool. Cut into strips width ways.
Lay a folded dishcloth down on your kitchen side, with the shortest side facing you and a large piece of cling film over the top. Place a nori sheet down on the edge closest to you, with the widest side closest to you.
Spread ¼ of the rice out on the nori sheet. Use wet hands to pat it down so you have an even rectangle of rice. Sprinkle over ¼ of the sesame seeds then, very carefully, flip over the sheet so that the rice is now underneath and the exposed nori is on top.
Spread 1 tbsp mayo on the nori sheet. Add a sprinkle of grated carrots, some cucumber sticks, a little pepper and a few bits of omelette to the bottom ⅓ of the nori. Roll the nori and rice over with your fingers and, once all the veggies are covered, roll over the clingfilm and towel, using it to mould and compress the sushi roll. Once all wrapped up, keep in the clingfilm and transfer to your fridge. Repeat with more clingfilm and the remaining ingredients until you have 4 rolls. Leave to chill in the fridge for approximately 30 minutes.
When you’re ready to serve, combine the avocado, wasabi and lime juice with a stick blender for 30 seconds until smooth. Divide up among 4 plates. Slice the sushi rolls and serve with the wasabi cream and soy sauce.