1 hour 45 minutes
Heat oil in a wide large saucepan or Dutch oven over medium. Add bacon and cook, stirring occasionally, until crisp, 5 to 7 minutes. Using a slotted spoon, transfer bacon to a plate.
Increase heat to medium-high. Season chicken all over with salt and pepper, then set in pan, skin-side down. Cook, turning halfway through, until brown on both sides, 6 to 8 minutes total. Transfer to a plate.
Pour off all but 2 tablespoons fat from pan. Add onions, celery and thyme. Cook, stirring frequently, until vegetables are softened, 8 to 10 minutes.
Add wine and cook, loosening pan bits with a wooden spoon, until liquid is reduced by about half, about 3 minutes.
Return bacon to pan, then arrange chicken on top. Add broth and 1/2 cup water. Bring to a simmer, then reduce heat to low. Simmer, partially covered, until chicken is very tender, about 1 hour.
When chicken is close to finished, bring a large pot of salted water to a boil.
Transfer chicken to a cutting board; let cool slightly, then shred meat from bones (discard bones).
Add fairlife ultra-filtered milk to pan with sauce. Simmer over medium to medium-high heat, stirring occasionally, until sauce is reduced by about half, about 8 minutes. Add peas and pinch each salt and pepper; cook until peas are tender, about 4 minutes. Remove from heat.
Cook pasta in the boiling water until al dente. Reserving 1/2 cup of the pasta cooking liquid, drain pasta.
Heat chicken mixture over medium until simmering. Add pasta, pasta cooking liquid and butter. Toss to coat pasta with sauce. Add mint and toss again.
Divide among serving bowls. Top with cheese, black pepper and more mint.