Fill a large pot halfway with hot water; add 1/2 teaspoon salt, the fennel wedges and carrots. Bring to a boil; cook until the vegetables are crisp-tender, 3 minutes. Remove with a slotted spoon. Reduce the heat to low.
Add the garlic, scallions, fennel fronds and chicken to the pot and gently simmer until the chicken is cooked through, 10 to 15 minutes. Fill a bowl with salted ice water. Transfer the chicken to the ice water using tongs; let sit 30 seconds, then remove to a cutting board.
Using a slotted spoon, transfer the garlic, scallions and fennel fronds to a blender; add 1/3 cup of the cooking liquid. Add the green beans to the pot, remove from the heat and let sit 5 minutes; drain.
Add the olive oil, vinegar, mustard, parsley and basil to the blender and puree; season with pepper. Cut the chicken into pieces and serve with the vegetables; drizzle with the herb sauce.