Place carrots and broccoli at the bottom of the prepared slow cooker.
Season chicken with salt and pepper and place on top of the vegetables.
In a medium bowl, whisk together condensed mushroom soup, condensed chicken soup, and water. Pour sauce over the chicken.
Sprinkle dry stuffing mix over top of the chicken. Drizzle with melted butter.
Put a lid on the slow cooker and cook on LOW heat for 4-6 hours or on HIGH for 2-3 hours.
Notes
The broccoli gets quite soft over the long cooking time, so feel free to omit that ingredient or replace it with a different vegetable if you don't care for super-tender broccoli florets. You can also just omit the veggies altogether, and instead cook the chicken and stuffing on their own.
While chicken breasts or chicken thighs will both work in this recipe, I find that the thigh meat stays more tender and juicy in the slow cooker.
Every slow cooker runs at a slightly different temperature (which is why the recipe gives you general time ranges, rather than exact times). Since my slow cooker tends to cook at a relatively high temperature, I find that the meal is best when cooked on HIGH for about 2 hours or on LOW for about 4 hours. Just get familiar with your pot and adjust accordingly. You'll know the dish is done when the chicken reaches an internal temperature of 165°F.
Do not prepare the stuffing mix according to the package instructions. The liquid condensation from the pot will mix with the stuffing and melted butter as it cooks to create a moist, buttery stuffing in the end.