Heat the oven to 350°F.

• Cut a small corner from the beef packaging and drain off any excess liquid. Transfer to a plate; pat dry with a paper towel.

In a large frying pan over medium-high heat, warm 1 tablespoon oil until hot but not smoking. Add the beef and Sloppy Joe seasoning blend, season with salt and pepper, and cook, stirring to break up the meat, until the beef is browned and cooked through, 5 to 7 minutes. Spoon off all but about 1 teaspoon fat from the beef. Stir in the green chiles and diced tomatoes and cook until slightly thickened, 1 to 2 minutes. If you prefer a saucier mixture, stir in 2 to 4 tablespoons water. Season to taste with salt and pepper. Remove from the heat.
While the beef cooks, prepare the buns.

On a sheet pan, lay the buns, cut sides up, and place 1 slice cheddar cheese on each bun bottom. Toast in the oven until the cheese melts, 4 to 6 minutes.
While the buns toast, prepare the salad.

• Juice the limes.
• Cut away any core from the cabbage; thinly slice the cabbage.
• Trim the ends from the radishes. Cut them in half, then cut the halves into thin half-moons.
• Peel the cucumber, if desired, and trim off the ends; cut the cucumber in half lengthwise, then crosswise into ¼-inch-thick half-moons.
• Cut the avocado in half lengthwise. Remove the pit, scoop out the flesh, and cut the flesh into ¼-inch-thick slices.
• Coarsely chop the cilantro.

In a large bowl, stir together 1 tablespoon oil and 2 to 3 tablespoons lime juice; season with salt and pepper. Add the cabbage, radishes, cucumber, avocado, and cilantro and toss to coat. Season to taste with salt and pepper.

Place the bun bottoms, cut sides up, on individual plates. Pile the beef mixture on top and garnish with as many pickled jalapeños as you like, then close with the bun tops. Serve the cucumber-radish salad on the side.