1. Boil sweet potatoes
Cut sweet potatoes into 1-inch pieces. Place in a medium saucepan with 1 whole garlic clove and enough cold salted water to cover by 1 inch. Bring to a boil over high heat, then reduce heat to medium and simmer until tender when pierced with a knife, 8–10 minutes. Reserve ¼ cup cooking water, then drain potatoes and return to saucepan. Cover to keep warm until step 6.
2. Cook veggies
Finely chop onion. Finely chop mushrooms.Trim ends from green beans, then cut crosswise into 1-inch pieces. Heat 1 tablespoon oil in a medium ovenproof skillet over medium-high. Add chopped onions; cook, stirring, until softened, 4–5 minutes. add mushrooms cook 4-5 min. Add green beans; cook, covered, until tender, about 5 minutes more. Stir in corn. Transfer veggies to a plate.
3. Cook turkey & season
Preheat broiler with a rack in the upper third. Heat 2 teaspoons oil in same skillet over medium-high. Add turkey; cook, breaking up into large pieces with a spoon, until browned and cooked through, 3–4 minutes. Sprinkle 2 teaspoons flour over turkey; stir to combine. Add chicken broth, 1 tablespoon Worcestershire, and 1–2 whole thyme sprigs; stir to combine.
4. Finish filling
Bring turkey mixture to a simmer over medium heat. Reduce heat to medium-low and simmer, stirring, until thick enough to coat the back of a spoon, 2–3 minutes. Stir in veggies and season to taste with salt and pepper.