Preheat oven to 350°. Heat 1 tbsp olive oil in a Dutch oven over medium heat. Pat chicken dry with a paper towel and season on both sides with salt and pepper. Add chicken to the pot skin-side down and cook for 4-5 minutes or until skin is lightly browned. Remove to a plate.
Add garlic to the oil that remains in the Dutch oven and cook briefly until fragrant and lightly browning on the edges. Add tomatoes, zucchini, thyme, bay leaf, red pepper flakes, and ½ tsp salt and cook for 2-3 minutes until tomatoes and zucchini start to soften. Return chicken to pot, skin-side up. Cover and transfer to oven for 1 hour, stirring sauce every 20 minutes.
During the final 20 minutes of braising, bring a large pot of salted water to boil. Add pasta and cook per package directions for al dente. Drain and return to pot
When done transfer chicken thigh to a cutting board. Remove bay leaf. Remove skin and bone from chicken and shred the meat with two forks. Return to pot and mix with sauce. Divide pasta into bowls and top with sauce. Garnish with feta and herbs.