Preheat the oven to 350 degrees F. Preheat the grill to medium-high heat.
For the garlic butter: Place the garlic in a small saute pan and set over medium heat. Add the butter and warm through. Cook the garlic until fragrant, 4 to 5 minutes. Then place the garlic butter in the fridge to firm up again.
For the blistered chile: Toss the poblano with olive oil to coat. Place the pepper under the broiler or char on the grill over a direct flame until the skin blisters. When done, remove the seeds, stem and roughly chop. Reserve.
For the onions and peppers: Set a large saute pan over high heat. Add a drizzle of olive oil and saute the onions and bell peppers until wilted, 5 to 6 minutes. Season with the paprika and some salt and pepper.
For the turkey burgers: In a large mixing bowl, combine the turkey, mustard, Worcestershire, cumin, white pepper and the reserved blistered chile. Sprinkle with kosher salt. Mix gently until well combined. Form into four 4-ounce burger patties. Wipe the grill down with oil-blotted towels, and then place the burgers on the hot grill and cook for 3 to 4 minutes. Flip the burgers, top with a slice of cheese and cook until cooked through and cheese has melted, 3 to 4 minutes. Set aside and allow to rest.
Smear the buns with the garlic butter. Grill the buns until golden and crispy, 20 to 30 seconds on each side.
To build the burgers, place 1 grilled turkey burger on the bottom half of the bun. Then top with a good heaping of sauteed onions and peppers. Finish with the top half of the bun and skewer to hold all of it together. Serve with extra mustard on the side.