Heat the oven to 400°F and arrange a rack in the middle. Rub the chicken on both sides with the olive oil and season generously on both sides with salt and pepper. Place on a baking sheet and roast until an instant-read thermometer registers 160°F, about 45 to 50 minutes.
Meanwhile, place the porcinis in a medium bowl, cover with the boiling water, and let sit for 20 minutes. While the mushrooms soak, finely chop the parsley leaves and tarragon leaves (you will need 1 tablespoon of each) and set aside together in a small bowl. Slice the cremini mushrooms 1/4 inch thick and set aside in a medium bowl. Finely chop the shallots, place in a small bowl, and set aside. Place the chicken broth or stock in a medium bowl and set it aside.
When the porcini mushrooms are ready, remove them from the water using a slotted spoon and transfer to a cutting board. Coarsely chop and transfer to a small bowl; set aside. Remove 1/2 cup of the mushroom soaking liquid (be careful not to include the gritty sediment on the bottom of the bowl) and add it to the chicken stock; set aside. Discard the remaining mushroom liquid.
When the chicken is ready, transfer it to a cutting board to cool. Meanwhile, heat the butter in a large frying pan over medium heat until foaming. Add the chopped shallot and cook, stirring occasionally, until softened, about 3 minutes. Add the cremini mushrooms and cook, stirring occasionally, until softened, about 5 minutes. Add the measured salt and pepper and the chopped porcinis and stir to combine.
Sprinkle in the flour and stir to coat the mushroom mixture. Cook, stirring often, until the raw taste has cooked out of the flour, about 3 minutes. Slowly whisk in the chicken stock mixture, whisking out any lumps. Add the milk and cream, increase the heat to medium high, and bring to a boil, whisking occasionally. Reduce the heat to low and simmer, whisking occasionally, until the sauce is smooth and thickened, about 10 to 15 minutes Meanwhile, remove the chicken meat from the bone and tear it into bite-sized pieces, discarding the skin, cartilage, and fat. Set the meat aside on the cutting board.
When the mushroom sauce is ready, stir in the chicken, peas, sherry, and chopped tarragon and parsley and cook until warmed through, about 5 minutes. Taste and season with salt and pepper as needed. Serve immediately over rice, pasta, toast, or puff pastry.
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