Halve the courgette and carrot lengthwise, then thinly slice. Peel and mince the garlic. Lay chicken flat, place your hand on top and slice horizontally along the middle, making two thinner fillets.
Cook chicken
Heat the oil in a large frypan or skillet over medium-high heat. Season the chicken with salt and pepper. Cook the chicken for 3-4 mins each side until golden and cooked through. Remove from the pan, leaving the oil in the pan on heat.
Cook veggies & pasta
Add more oil to the pan for the vegetables if needed. Cook the courgette, carrot and garlic, stirring occasionally, for 2 mins, or until the carrot is slightly softened. Then add the stock and milk, bring to a simmer, then add the pasta. Move pasta around every 30 seconds until it is softened (around 3 mins) so it doesn't stick. Once pasta is softened, reduce heat to medium and stir every few minutes.
Make sauce
At about 9-10 mins, the pasta should be almost cooked but there should still be liquid covering the base of the pan. Add coconut cream and grate in the parmesan (reserving a small amount for garnishing) and mix. Simmer, stirring occasionally, for 2 mins, until the sauce is thickened and the pasta is cooked. (If you cook it too long and the sauce goes gluggy, add a splash of hot tap water - will go back to silky with a few tosses).
Finish alfredo
Slice the chicken. Add the chicken, with any resting juices, and spinach to the pasta mixture and gently stir to combine. Taste, then season with salt and pepper.
To serve
Divide the chicken & spring veggie alfredo pasta between plates. Sprinkle over the remaining grated cheese.