Melt 1 tablespoon butter in a large saucepan over medium heat. Add potato, onion, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Cover and cook 6 minutes. Add broth and 2 cups water; bring to a boil. Cook 10 minutes or until potato is tender.
Combine 3 tablespoons water and remaining 1 tablespoon butter in a large Dutch oven; bring to a simmer. Add cabbage and thyme. Cover and cook 5 minutes, stirring occasionally. Remove from heat; stir in remaining 1/4 teaspoon salt and remaining 1/4 teaspoon pepper.
Place half of potato mixture in blender. Remove center piece of blender lid (to allow steam to escape); secure blender lid on blender. Place a clean towel over opening in blender lid (to avoid splatters). Blend until smooth. Pour into a large bowl. Repeat procedure with remaining potato mixture. Add potato mixture to cabbage mixture; cook over medium-low heat until thoroughly heated.