Preheat oven to 425°F. Toss carrots, chopped Granny Smith apples, and thyme with 2 Tbs. oil and 1/2 tsp. salt on a parchment-paper-lined sheet pan. Roast in an even layer until carrots are tender, 30 to 35 minutes.
Meanwhile, heat remaining 1 Tbs. oil in a large pot over medium heat. Add onion and cook until softened, about 5 minutes. Add garlic and cook 1 minute more.
Add apple cider and water to pot and bring to a simmer. When carrot mixture is done roasting, add to pot. (Keep oven on.) Simmer, covered, until carrots are very soft, about 10 minutes. Remove thyme.
Blend soup until smooth using a handheld immersion blender (or carefully transfer to a regular blender). Add vinegar, 2 Tbs. yogurt, remaining 1/2 tsp. salt, and pepper to taste. Stir in extra water or broth as needed to get desired consistency.
Toast bread lightly on a sheet pan in oven, about 2 minutes. Remove pan; sprinkle slices with three quarters of cheese. Top each with 5 to 6 Gala apple slices, and sprinkle remaining cheese on top. Toast in oven until melted, about 5 minutes. Cut in half.
Ladle soup into bowls; top with a dollop of additional yogurt, if using. Serve with toasts.
