Combine 2 tablespoons brown sugar, the instant coffee, cocoa powder, mustard powder, chile powder, cinnamon and 1 teaspoon salt in a bowl; rub the mixture between your fingers until fine. Season the steak with salt, then generously rub with the coffee-spice mixture.
Heat the vegetable oil in a large cast-iron skillet over medium heat. Working in batches if necessary, sear the steak, 3 to 6 minutes per side for medium rare. Transfer to a cutting board and let rest; reserve the juices in the skillet.
Add the remaining 1 teaspoon brown sugar and the onion to the skillet. Sprinkle with 1/4 teaspoon salt, and pepper to taste; cook over medium-high heat, stirring frequently, until the onion is soft and golden, about 5 minutes. Add the bell pepper and 1/4 cup water; cook, stirring, until crisp-tender, about 5 minutes. Stir in the lime juice. Season with salt and pepper.
Slice the steak against the grain. Divide the steak, bell pepper, onion and the juices from the skillet among plates. Serve with lime wedges and cornbread.