In a large bowl, combine the ground turkey, jerk seasoning, sliced green onions, minced garlic, salt, and pepper. Mix gently with your hands until just combined; do not overwork.

Divide the mixture into 4 equal portions. Shape each into a patty about 3/4 inch thick. Press a slight indent in the center of each patty to help them cook evenly.

Preheat a grill or cast-iron skillet over medium-high heat. Lightly brush the grates or pan with olive oil.

Place the patties on the grill and cook for 5 to 6 minutes per side, or until an instant-read thermometer inserted into the center registers 165°F (74°C). If adding cheese, lay a slice on each patty during the last minute of cooking.

In a medium bowl, toss together the shredded cabbage, chopped jalapeño, cilantro, lime juice, olive oil, and salt. Taste and adjust seasoning if desired.

While the patties rest, lightly toast the burger buns on the grill or in a toaster.

Place each patty on a bun bottom. Top with a generous spoonful of spicy jalapeño slaw and a few slices of avocado. Add the bun top and serve immediately with any extra slaw on the side.