Combine the crushed tomatoes, vegetable broth, vodka, Worcestershire sauce, hot sauce, celery salt, horseradish, and lemon juice in a large pot. Stir well and bring to a simmer over medium heat. Cook for 15 minutes, stirring occasionally, until the flavors blend.
While the soup simmers, assemble the sandwiches. Place two slices of cheddar cheese between two pieces of bread to make four sandwiches total.
Heat a large skillet over medium heat. Spread butter on the outside of each sandwich. Cook two sandwiches at a time for 3 to 4 minutes per side until the bread is golden and the cheese melts.
Remove the sandwiches from the skillet and cut each one into four strips to create dippers.
Ladle the soup into bowls and serve with the grilled cheese dippers and a celery stick on the side.