Chop one head of broccoli into small broccoli florets to yield about 3 cups. Finely chop 2 - 3 cloves of garlic.
Heat 2 tablespoons of olive oil in a cast iron skillet over medium heat. Sprinkle with salt and pepper for seasoning. Cook broccoli florets until they are softened and begin to char (about 15 - 20 minutes), stirring occasionally. Add garlic and continue cooking another 3 - 4 minutes. Transfer broccoli and garlic mixture to a plate, set aside and wipe the skillet to remove debris.
While broccoli is cooking, drain the sauerkraut with a fine mesh sieve. In another frying pan, melt ½ tablespoon of butter and add drained sauerkraut. Cook about 5 - 10 minutes on medium heat until sauerkraut is warmed through.
Butter 1 side of the 4 slices of marble rye bread. On the non-buttered side, spread about 1 tablespoon of Russian dressing. Place one slice of Swiss cheese over the Russian dressing.
In the cast iron skillet, place one slice of rye, buttered side down. Top with 1/2 of the charred broccoli and garlic to create a layer. Top with about 1 - 3 tablespoons of sauerkraut to cover the broccoli. Finally, top with a second slice of rye, with the buttered side facing up.
Grill the broccoli Reuben, pressing the sandwich with a spatula to flatten. until the bottom is golden brown and toasted, about 4 minutes. Then, carefully flip over and grill the other side. (Repeat the previous step and this step to make the second broccoli Reuben.)
Cut on a diagonal and serve immediately warm.