Prep - Set oven rack to bottom. Preheat oven to 300°F (150°C). Cover the dried mushrooms with boiling water and leave 15 minutes to soften. Toast the pine nuts in the oven until golden, about 10 minutes. Remove and set aside. Leave oven on.
Cook Chicken - Rinse chicken breast a pat dry with paper towels. Heat a cast-iron pan over medium-hot heat. Season breasts all over aggressively with salt and place ⅓ of the butter and then the breasts in the pan skin-side down. Cook until skin turns golden, about 3 minutes. (Keep breasts skin-side down and don't turn them over at any time). Tilt skillet and remove excess fat. Add shallots, lemon juice, and vermouth. Continue cooking over moderate heat until juices are reduced to a glaze. Add ¼ cup demi-glace to the skillet, cover and put back into oven for another 12 - 15 minutes. Chicken should be just firm to the touch
Make Mushroom Mixture - While chicken is in the oven, strain mushroom soaking water through a coffee filter into a measuring cup. Carefully wash mushrooms again to be sure there's no grit. Dry in a colander. Heat 2 tbsp olive oil in a 10" skillet over medium heat. Add fresh and drained mushroom. Sauté for 5 minutes. Add reserved soaking liquid from mushrooms and reduce to a glaze, about 10 minutes. Stir in garlic, parsley, and salt to taste. Transfer to a bowl. Add ham cubes to the same skillet, cover and cook over low heat for 2-3 minutes. Mix mushroom mixture back into the pan and remove from heat.
Finish Cooking Dish - Transfer breast to a cutting board. With a thin knife, separate meat from the bone in one piece. If breasts aren't fully cooked, return to skillet with cooking liquid and poach gently for 1 to 2 minutes. Place breasts skin side up on a cutting board and slice each into 4 diagonal pieces. Skim off the fat from the cooking liquid in the skillet and add remaining demi-glaze. Place skillet over high heat and reduce the liquid by half, about 5 minutes. Remove from heat and swirl in the remaining butter 1 tbsp at a time. Add Cognac and salt and pepper.
Serve - Overlap chicken slices on a serving platter or on individual plates. Strain sauce over the chicken. Spoon reheated mushroom mixture next to chicken and sprinkle chicken slices with pine nuts. Enjoy.