Preheat the oven to 375 degrees F.
Heat a non-stick pan on medium heat. Once hot, add the oil. Sauté the onion, celery, carrots, and garlic until the celery starts to soften. Add the chicken, sage, and thyme; brown while breaking it up into smaller pieces. Stir in the flour until fully combined. Season with salt and pepper. Stir in the broth and kitchen bouquet. Simmer until the liquid starts to thicken and concentrates in flavor. Taste and adjust the seasoning with salt. Stir in the peas.
Toss the potato slices with a little oil, and season well with salt and pepper.
Pour the mixture into an 8 x 8 baking dish (or larger if batch cooking portions). Stick the potato slices in a line on top of the mixture with sliced edges facing somewhat upwards. Continue doing this until the surface of the mixture is covered, using smaller pieces of potato to fill in any gaps. Sprinkle with parmesan cheese over top.
Place in the oven and bake until the potatoes are tender and golden brown, about 40 to 50 minutes.
Remove from the oven and let set for about 5 minutes before serving.