- Preheat oven to 350 degrees. Season chicken liberally with salt and pepper. Set cast iron pan or Dutch oven over medium-high heat, then add olive oil. Sear chicken, skin side down, until golden brown, 3–4 minutes. Flip chicken and cook for 1 minute. Remove chicken from pan and set aside.
- Add onion, celery, garlic, bay leaves, and thyme to still-warm pan. Cook until tender. Add white wine and cook until reduced by half. Add chicken back to pan on top of vegetables and add chicken stock. Put in oven and cook 1½ hours.
- When chicken is cooked, let cool at room temperature, then remove chicken and strain vegetables from liquid. Place liquid in refrigerator and discard solids.
- When liquid is cool, skim off fat and discard. Pick chicken meat off bone and discard bones.
- Preheat oven to 350 degrees. Rub potatoes with olive oil, salt, and pepper. Poke holes in potatoes with fork and wrap each potato completely in foil. Bake until very tender, approximately 1 hour.
- When potatoes are cooked, remove foil. When cool enough to handle, remove flesh and mash until smooth. Add cheese, salt, nutmeg, and cayenne. Mix well.
- Slowly add flour and mix until doughy consistency is reached. Reserve remaining flour.
- To roll dumplings, spread flour on smooth work surface. Using both hands, roll dough into long cylinder, starting with hands close together and slowly spreading them apart.
- Cut cylinder into 1-inch dumplings. Store on parchment paper dusted with flour until ready to use.
- Place all ingredients except banana peppers in large pot. Bring to boil, then reduce to simmer. Simmer 5 minutes. Remove from heat.
- Place banana peppers in bowl and cover with liquid. Place in refrigerator. Pickles are ready to use when they are cool.
- Bring pot of salted water to boil. In a sauté pan, add chicken meat and reserved chicken stock. Bring to boil, then reduce to simmer. When liquid is reduced by 90 percent, add butter and stir to combine. Remove from heat.
- Add dumplings to boiling water. They will float when cooked. Remove with strainer and add to chicken mixture. Toss to combine. Place mixture in serving bowl. Top with blue cheese crumbles and pickled peppers. Serve warm.