Heat a large Dutch oven over medium heat. Add 2 tablespoons oil. Add onions; cover and cook 12 minutes or until tender, stirring occasionally. Stir in 2 teaspoons thyme; cook, uncovered, 1 hour or until deep golden brown, stirring occasionally.
Stir in flour; cook 2 minutes, stirring constantly. Stir in brandy; cook 1 minute. Stir in wine; cook 1 minute. Place onion mixture in a large bowl.
Return pan to medium heat. Add remaining 2 teaspoons oil; swirl to coat. Add celery, carrot, mushrooms, and bay leaf; cook 5 minutes, stirring occasionally. Add stock, Worcestershire, and parsley. Bring to a boil; reduce heat to medium, and simmer 15 minutes. Remove from heat; let stand 10 minutes. Strain broth through a fine sieve over a bowl; discard solids. Return broth to pan; add onion mixture, liquid aminos, salt, and pepper. Bring to a simmer; cook 15 minutes.
Preheat broiler to high.
Arrange bread slices in a single layer on a baking sheet; broil 1 minute or until tops are golden. Ladle 1 cup soup into each of 8 broiler-safe soup crocks. Top each serving with 2 bread slices, toasted side down. Sprinkle evenly with cheese. Working in batches if needed, place crocks on baking sheet, and broil 1 minute or until tops are golden and cheese bubbles. Garnish with remaining thyme.