For the mini Reubens: Preheat the broiler. Fit a rack 8 to 6 inches from the heat source.
Brush the melted butter over each bread slice. Flip over and add a layer of Russian dressing to each slice, followed by a piece of cheese. Layer the corned beef on half the bread slices, followed by a spoonful of sauerkraut. Build 3 sandwiches with the remaining slices, cheese-side down.
Broil, flipping halfway through, until brown and toasty, about 4 minutes. (Keep an eye on this so that it doesn't burn.) Cut off the crusts and slice each sandwich into 4 squares. Place a cornichon, the sandwich and another cornichon on a skewer. Repeat to make 5 more skewers.
For the Bloody Marys: Mix together the tomato juice, vodka, brine, hot sauce, lime juice, celery salt, Worcestershire sauce, caraway seeds, horseradish, sugar, garlic and a few turns of black pepper in a pitcher. Taste and adjust the seasonings as desired. Garnish with celery stalks and the Reuben skewers. Serve the remaining mini Reubens on the side.