How to Make It
Preheat oven to 375°F. Finely chop 1 ounce bread, and set aside. Cut remaining 4 ounces bread into 16 sticks, or soldiers. Stir together butter and 1 tablespoon oil in a small bowl; brush evenly over all sides of breadsticks. Set aside any remaining butter-oil mixture. Arrange breadsticks in a single layer on an aluminum foil–lined baking sheet. Bake at 375°F, turning occasionally, until evenly toasted, about 12 minutes.
Meanwhile, heat remaining 2 tablespoons oil and any remaining butter-oil mixture in a large saucepan over medium-low. Add shallots, garlic, and carrot. Cover and cook, stirring occasionally, until vegetables are tender when pierced with a fork, 10 to 15 minutes. Increase heat to medium-high. Add wine; cook until liquid is mostly evaporated, about 2 minutes. Add chopped bread, stock, salt, and thyme; bring to a boil. Reduce heat to medium-low; simmer, uncovered, until vegetables are very tender and bread has dissolved, about 10 minutes.
Discard thyme sprigs. Pour chicken stock mixture into a blender; add half-and-half. Remove center piece of blender lid (to allow steam to escape); secure lid on blender. Place a clean towel over opening in blender lid (to avoid splatters). Process until very smooth, about 1 minute. Return chicken stock mixture to pan over medium, and cook, stirring occasionally, until warmed through, about 2 minutes. Ladle soup into each of 4 bowls; garnish with pepper, if desired. Serve with breadsticks.