Heat a large skillet over medium-high heat and add the oil. Cook the beef, onion and garlic until browned, about 8 minutes. Remove from the heat and drain off any excess fat. Mix in the ketchup, Worcestershire, hot sauce and some salt and pepper. Stir in the processed cheese and pepper jack until melted.
Put the rye slices on a baking sheet and add a small scoop of filling to each one. Pat down slightly to stick it to the bread, then flash-freeze for 45 minutes. Store in freezer in resealable plastic bags until using.
To bake, preheat the oven to 350 degrees F. Bake from frozen state until the cheese is melted and bubbly, about 25 minutes. Sprinkle with green onions before serving.