Remove from heat; cool 5 minutes. Place half of soup mixture in blender; process until smooth. Return mixture to pan. Repeat procedure with remaining soup mixture.
Add half-and-half and heavy cream, stirring with a whisk over low heat. Set aside; keep warm.
While soup simmers, place 3 bread slices on a work surface; arrange 2 cheese slices on each of 3 bread slices. Top with remaining 3 bread slices. Spread softened butter on sides of sandwiches. Heat a large nonstick skillet over medium heat. Add sandwiches to pan; cook 4 minutes or until lightly browned. Turn sandwiches over; cook 2 minutes or until cheese melts. Cut each sandwich in half diagonally.
Ladle soup into bowls; serve with sandwiches.
Notes
Just like Mom used to make, only better, because the soup doesn't come out of a can, which means you can control the sodium. Keep the sandwich old-school with American cheese.