Heat oil in large saucepan over medium-high heat. Add onion and carrot; cook 2 to 3 minutes or until onion is tender. Add garlic; cook 1 minute more or until fragrant.
Add water, tomato sauce and tomato paste to saucepan; stir to combine. Reduce heat and simmer 15 minutes or until thickened slightly, stirring occasionally.
Remove soup from heat. Carefully blend mixture in saucepan with hand-held immersion blender until smooth. Stir in milk and keep warm; set aside.
Make grilled cheese croutons by buttering 1 side of each bread slice. Place Cheddar cheese on half the bread on the non-buttered side. Top with remaining bread slices, buttered side out. Cook in large skillet over medium heat until both sides are golden brown and cheese is melted. Let sandwiches stand 2 minutes. Trim top crust and cut into cubes. Place half of a cubed sandwich into each bowl of soup. Sprinkle with Parmesan cheese; serve.