Heat your oven to its lowest setting. Peel and grate 1 garlic clove into a small bowl. Crack in the eggs. Pour in 1 tsp tamari. Beat together and set aside.
Peel and grate the remaining garlic clove and the ginger. Mush together in a medium bowl. Add half the remaining tamari to the bowl. Add the beef strips and toss to coat. Put to one side to marinate.
Peel and thinly slice the red onion. Peel and slice the carrots into matchsticks. Thoroughly wash and dry the bean sprouts. Halve the chilli and flick out most of the seeds. Finely chop it.
Tip 1 tbsp cornflour into a bowl. Lift the beef stir-fry strips from the marinade and shake off any excess liquid. Place them into the bowl with the cornflour and toss to coat. Reserve the marinade. Add the honey to the reserved marinade and whisk well.
Heat a frying pan for a couple of mins. When warm, drizzle in ½ tbsp oil. Pour in the egg mix and swirl to coat the pan. Cook for 2 mins, till set on the top. Slide onto a plate and place into the oven to keep warm.
Wipe the pan and place it back on the heat. When warm, drizzle in 1 tbsp olive oil. Add the beef and fry for 2 mins till golden. Scatter in the sesame seeds and toss to coat the beef in the seeds. Lift the strips out on to a clean plate and cover with foil.
Keep the pan on the heat. Add ½ tbsp oil. Throw in the carrots, onion, chilli and bean sprouts and stir-fry for 3-4 mins till the onion has slightly softened. Add the spinach and stir to wilt it. Pour in the marinade you reserved earlier and bring to a bubble.
Spoon the veg out onto a couple of warm plates. Top with the sesame beef. Roll up the omelette and thinly slice it. Serve on the side of the sesame beef and gingery veg. Squeeze wedges of lime over to taste.