Combine mushrooms, shallots, oil, and 1/4 cup water in a 5- to 6-quart Dutch oven. Cover and cook over high heat, stirring often, until mushrooms are juicy, 3 to 4 minutes. Uncover and cook, stirring often, until the mushrooms are lightly browned, 8 to 10 minutes. Add broth, carrots, thyme, and bay leaves; bring to a boil.

Place chicken in a 5- to 6-quart slow cooker and lay lemon slices on top. Turn heat to high. Carefully pour in the vegetable mixture. Cover and cook until the chicken is very tender, 3 1/2 to 4 hours.

With a slotted spoon, transfer the chicken and vegetables to a bowl; discard bay leaves and lemon slices. Skim fat and pour juices into a large saucepan; add lemon zest. Bring to a boil over high heat. Boil until reduced to 2 cups, 15 to 20 minutes.

Mix cornstarch with remaining 1/4 cup water in a small bowl. Add to the pan and cook, stirring, until slightly thickened. Add cream and lemon juice; stir until boiling. Return the chicken and vegetables to the sauce and heat through. Season with salt and pepper. Just before serving, stir in peas and parsley.