Combine the olive oil, garlic and parsley in a mixing bowl and whisk until smooth.
Preheat the oven to 180°C.
Line a baking tray with nonstick baking paper and sprinkle the grated Parmesan onto it in roughly rounds, spaced apart. Bake for 2-3 minutes until just set but not coloured too much. Remove and allow to cool on the trays for a few minutes before carefully lifting onto a wire rack to cool and harden.
Heat a frying pan over a moderate heat until hot, then add half of the olive oil dressing and stir-fry the chicken for 4-5 minutes until just cooked with no pink showing. Remove from the heat and set to one side.
When you are ready to assemble the dish, sprinkle a few slices of red onion onto the base of serving plates.
Layer a Parmesan crisp, then top with spinach, rocket, cherry tomatoes and pieces of chicken, trying to alternate the ingredients of the mille-feuille as evenly as possible in a tower in the centre of the plates. Spoon over the remaining dressing and serve immediately.
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