Season meat with salt and pepper. Set aside.

Melt 1/3 stick of butter in a heavy skillet over medium-high heat. Cook onions for several minutes, until golden brown. Remove to a separate plate and set aside.

Return skillet to stove and increase heat to high. Add another 1/3 stick of butter. When melted, add meat to pan in a single layer. Do not stir. Allow meat to sit, undisturbed, for a minute in order to brown. With a spatula, flip meat to the other side and allow to brown. Reduce heat to low. Add onions back into pan, then add Worcestershire, Tabasco, and remaining 1/3 stick of butter. Stir and allow to simmer, adding more Worcestershire or Tabasco as desired.

Melt 1/2 stick of butter on griddle or in a separate skillet. Place roll halves, cut side down in butter and brown over medium-low heat until golden brown.

To serve, spoon meat/onion mixture over bottom half of rolls. Spoon 1-2 tablespoons pan juice over the top. Lay two mozzarella slices on each sandwich, followed by two tomato slices. Top with a generous amount of arugula, then press the top half of the roll on top. Cut in half and serve immediately.