Heat 1 teaspoon oil in a large nonstick skillet over medium-high heat. Add tomatoes; cook 4 minutes, stirring once. Remove from heat; stir in 2 tablespoons vinegar. Sprinkle tomatoes with thyme, 1/4 teaspoon salt, and pepper. Wrap handle of pan with foil; bake at 300° for 15 minutes. Keep warm.
Combine mayonnaise, mustard, and garlic in a small bowl. Spread mayonnaise mixture evenly over top half of bread loaf. Spoon tomatoes evenly over bottom half of loaf. Arrange Brie over tomatoes; top with chicken. Combine 2 teaspoons oil, 1 teaspoon vinegar, and 1/8 teaspoon salt in a medium bowl, stirring with a whisk. Add spinach, tossing gently to coat. Top chicken with spinach mixture; replace top half of bread. Cut loaf into 6 pieces.