Season the chicken with Chicken seasoning, smoked paprika, and ½ teaspoon black pepper.
Heat the olive oil and 1 tablespoon butter in a large pot over medium-high heat. Cook the chicken for about 8–10 minutes total, turning as needed, until the thickest part reaches 165°F (74°C). Remove and set aside. Chop or shred it.
Add the remaining 4 tablespoons butter, onion, carrots, and celery. Season with salt and pepper and cook for about 10 minutes, stirring occasionally, until softened. Adjust the heat as needed to prevent burning.
Add the garlic and cook for 1 minute. Stir in the flour, bouillon powder, and herbes de Provence. Cook for 2–3 minutes, stirring frequently.
Slowly stir in the water, scraping the bottom of the pot, then add the evaporated milk. Bring to a gentle boil.
Shred or chop the chicken and return it and any juices to the pot. Reduce the heat to low and simmer for 10 minutes.
Meanwhile, cut each biscuit into nine roughly equal pieces.
Bring the chicken mixture back to a gentle boil. Add the biscuit pieces individually around the pot. Do not stir; gently press them into the liquid.
Cover and cook over medium-low heat at a gentle simmer for 15–20 minutes, or until the dumplings are cooked through. Check one from the center for a fluffy interior without raw, sticky dough; cook longer if needed.
Gently stir, add the parsley, and adjust salt and pepper to taste.