Step 1: Poach the Chicken

1. In a large pot, combine the chicken chunks, chicken broth, carrots, onion, celery, garlic, bay leaf, thyme, and black peppercorns.
2. Bring the mixture to a gentle simmer over medium heat. Avoid boiling, as this can toughen the chicken.
3. Once simmering, reduce the heat to low, cover the pot, and poach the chicken for 20-25 minutes, or until the chicken is fully cooked and tender.
4. Use a slotted spoon to remove the chicken and carrots from the pot. Strain the broth through a fine-mesh sieve into a bowl, discarding the solids. Reserve the broth for the sauce.

Step 2: Sauté the Mushrooms

1. In a large skillet, melt 1 tablespoon of butter over medium heat.
2. Add the halved or quartered mushrooms and sauté for 5-6 minutes, or until they release their moisture and begin to brown. Set the mushrooms aside.

Step 3: Make the Sauce

1. In the same skillet, melt the remaining 1 tablespoon of butter over medium heat. Stir in the flour and cook for 1-2 minutes, whisking constantly, to create a roux.
2. Gradually add 2 cups of the reserved chicken broth, whisking continuously to prevent lumps. Cook for 3-4 minutes, or until the sauce thickens slightly.
3. Stir in the heavy cream or half-and-half and season with salt, black pepper, and lemon juice. Let the sauce simmer gently for 2-3 minutes.

Step 4: Combine and Simmer

1. Return the poached chicken, carrots, and sautéed mushrooms to the skillet with the sauce. Stir gently to coat everything in the sauce.
2. Simmer the mixture over low heat for 5-7 minutes to allow the flavors to meld. Taste and adjust seasoning as needed.

Step 5: Serve and Garnish

1. Serve the Chicken Blanquette over rice, egg noodles, or mashed potatoes.
2. Garnish with freshly chopped parsley for a pop of color and freshness.