Preheat the oven to 450 degrees F. Combine the coleslaw mix, giardiniera and brine, 2 tablespoons olive oil and the sugar in a large bowl; toss well. Set aside at room temperature.
Heat the remaining 2 tablespoons olive oil in a large pot over medium heat. Add the garlic and cook until just golden, about 2 minutes. Add the red pepper flakes and chicken broth. Bring to a simmer and stir in the oregano; cook 2 to 3 minutes. Add the broccoli rabe, cover and cook until just tender, about 3 minutes.
Arrange the rolls cut-side up on a baking sheet and bake until just lightly toasted, 2 to 3 minutes. Remove and immediately top each with 2 slices cheese; let melt slightly.
Remove the broccoli rabe to a bowl using tongs and add the roast beef to the pot. Cover and simmer until hot, about 1 minute. Divide the roast beef among the rolls and top with the broccoli rabe. Serve with the slaw.