Pat 2 large skinless, boneless chicken breasts (about 1½ lb.) dry with paper towels. Season with kosher salt on both sides. Set aside.

Heat 3 Tbsp. extra-virgin olive oil in a medium Dutch oven over medium-high. Cook 10 oz. mixed mushrooms, trimmed if needed, torn into bite-size pieces, stirring occasionally, until softened, 5–7 minutes. Season with salt and continue cooking, stirring occasionally, until browned and crisp, 5–9 minutes more, depending on type. Transfer to a bowl.

Pour 1 Tbsp. extra-virgin olive oil into same pot, then add 3 large shallots, finely chopped, and 5 garlic cloves, thinly sliced, and season with salt. Cook, stirring occasionally, until softened and lightly browned, about 4 minutes. Add 1 Tbsp. all-purpose flour and 1½ tsp. dried thyme or 1 Tbsp. chopped thyme and cook, stirring often, until combined, about 1 minute.

Pour in 1 cup dry white wine, scraping up browned bits, and bring to a simmer. Cook until liquid is slightly thickened, about 5 minutes. Add half of the mushrooms, 1½ tsp. Diamond Crystal or 1 tsp. Morton kosher salt, reserved chicken, and 5 cups water; season with freshly ground pepper. Bring to a rapid boil. Cover pot; remove from heat. Let sit until chicken is cooked through, about 15 minutes.

Meanwhile, mix 3 Tbsp. finely chopped parsley, 2 garlic cloves, finely grated, remaining ¼ cup extra-virgin olive oil, and a big pinch of salt in a small bowl to combine.

Uncover pot and, using tongs, transfer chicken to a plate; shred with 2 forks. Return to pot and add 1 cup heavy cream and 2 tsp. Worcestershire sauce. Set over medium and heat until warmed through, about 4 minutes.

Ladle soup into bowls and top with remaining mushrooms. Drizzle with parsley oil; season with more pepper. Serve with country-style bread if desired.