Preheat a large, heavy-bottomed pot over high heat with EVOO, about two turns of the pan. While the EVOO is heating up, season the chicken with salt and freshly ground pepper.
Once the oil starts to ripple, add in the chicken breasts, seasoned side down, and brown them on both sides, about 4-5 minutes. Remove and reserve on a paper towel-lined platter. Add a little more EVOO to the pot and repeat with the thighs, reserving them on the same platter.
Once all the chicken is browned, add the carrots and onions to the pot, and a drizzle of EVOO if needed. Cook until the veggies start to get tender.
Tie the herbs together into a neat little bundle with some butcher twine and toss it in the pot. Season the veggies with salt and freshly ground black pepper then add in the 1/2 cup calvados. Remove the pan from the heat and, using a lighter or match, ignite the brandy to burn off the alcohol.
Once the flame has gone out, place the pot back over medium-high heat and add in the hard cider. Cook and reduce the liquid by 1/2, then add in the chicken stock, hot sauce, and nutmeg. Remove the herb bundle and puree the mixture in a food processor or a blender.
Return the liquid back to the pot, and add the herb bundle back in along with the browned chicken pieces. Cook over medium to low heat, simmering everything for about 15-20 minutes.
While the chicken is cooking, melt the butter in a skillet over high heat and cook the apples slices until lightly browned don't overcook them, they should still be crispy. Add in the remaining 2 tablespoons of Calvados, remove the pan from the heat and ignite the skillet like before. Let the alcohol cook out and reserve the apples for garnish for the creamy cider chicken.
Add the cream to the chicken and cook to warm through. Just before serving. add in a handful of roughly chopped parley. Serve with caramelized calvados apples on top and a chunk of multi-grain baguette alongside to mop up the sauce.