Preheat the oven grill (broiler) to high.
Heat the olive oil in a large, deep, heavy-based pan over medium heat.
Sprinkle the chicken with salt and pepper and cook for 5–6 minutes, turning once, until cooked through.
Remove the chicken and set aside; shred it once cool enough to handle.
Use the ¼ cup of water to deglaze the pan if needed, scraping up any sticky bits.
Add the butter, onion, and garlic to the pan and cook for 1–2 minutes, stirring.
Add the leek and cook, stirring, for 3–4 minutes until softened.
Add the flour and cook, stirring, for 1 minute until well incorporated.
Pour in the chicken stock and stir until combined.
Add the thickened cream, Italian mixed herbs, broccoli, salt, and pepper, and bring to a simmer.
Using tongs, add the cut lasagne squares, making sure they are completely submerged in the sauce.
Reduce the heat to low, cover, and simmer for 6–7 minutes to soften the pasta.
Remove the lid and continue to simmer, uncovered, for 4–6 minutes until the pasta is almost cooked and the sauce has thickened. Stir often to prevent sticking.
Add the shredded chicken and Parmesan to the pan; stir them through carefully.
Remove from the heat and sprinkle the mozzarella cheese over the top.
Place the pan under the grill and cook for 3–5 minutes until the cheese is golden brown and bubbly.
Allow to stand for 5 minutes before serving with fresh oregano or basil leaves.