In a large metal casserole dish, bring the chicken broth to a boil. Add chicken thighs, cover and simmer over medium-low heat for about 30 minutes or until the juices flowing from the chicken when pricked with a fork are clear. Using a skimmer, remove the chicken thighs from the broth and place them on a plate. Let cool. Boning the chicken thighs and cut the flesh into pieces. Reserve.
In the casserole dish containing the broth, add the carrots and potatoes. Cover and cook for 10 minutes. Add the pearl onions, cover and simmer for about 5 minutes or until they are tender. Using a skimmer, remove the vegetables from the broth and put them in a bowl. Reserve. Pour the broth into a large measuring cup (if necessary, add enough chicken broth to make 5 cups / 1.25 L of broth in total). Reserve.
In the casserole dish, melt 3 tablespoons (45 ml) of the butter over medium-high heat. Add celery, onion and mushrooms and cook, stirring often, for about 8 minutes or until the vegetables have softened. Add all-purpose flour, thyme, 1/2 teaspoon (2 mL) salt and pepper and cook, stirring, for 1 minute. Using a whisk, gradually add the reserved broth and bring to a boil, whisking. Reduce heat to medium-low and simmer, stirring often, for about 5 minutes or until the sauce has thickened enough to coat the back of a spoon. Return the reserved chicken and its cooking juices to the casserole dish. Add the reserved vegetables, peas and cream and mix.
In a bowl, using a whisk, combine the cake and pastry flour, parsley, baking powder and the rest of the salt. Add the rest of the butter and, using a pastry cutter or two knives, work the mixture until it has the texture of coarse breadcrumbs. Using a fork, add enough milk to make a sticky paste. Drop the dough into a ball in the simmering stew, 1 tablespoon (15 ml) at a time, leaving enough space between each. Cover and bake, without lifting the lid, for 15 minutes or until the dumplings are cooked inside.