Cook broccoli rabe in a large pot of boiling salted water, uncovered, until tender, about 4 minutes. Drain well in a colander, then chop.
Heat half of the oil in a heavy skillet over medium heat, then add garlic and cook, stirring, until garlic is fragrant and just begins to turn golden, about 1 minute.
Add broccoli rabe and cook, stirring frequently, 5 minutes.
Remove from heat and season to taste with salt and pepper.
Heat a panini or sandwich press according to manufacturer's instructions until hot. (Alternatively, heat a well-seasoned ridged grill pan over medium heat.)
Brush the bread on 1 side with the remaining oil.
Put slices, oiled sides down, on a work surface, then divide the cheese and soppressata evenly between 4 of the slices.
Spoon some of the broccoli rabe mixture over the soppressata.
Top with the remaining 4 bread slices, oiled sides up.
Put sandwiches on press, then pull down top onto sandwiches and cook until sandwiches are browned and crisp, 4 to 8 minutes. (If using a grill pan, put a heavy pan on top of sandwiches and cook, turning sandwiches over once.)
Remove from heat, cut sandwiches in half, and serve immediately.