In a skillet add 1 tablespoon olive oil and 1-2 tablespoon butter. Allow the butter to melt but not burn. Add the onion and sprinkle with a pinch of salt. Allow the onions to caramelize by stirring occasionally- until golden brown. Set aside and allow to cool down a bit.
Combine the turkey, garlic powder, cheyenne pepper, thyme and caramelized onions. Divide the mixture into 4 equal parts. Insert the cream cheese medallion in the middle of the burger and cover with meat- Shape each burger into roughly 1 inch circles. Sprinkle each burger with a pinch of salt and pepper.
Stove top option:
Heat oil in a non stick skillet/grill pan over medium -high heat. Add burgers to skillet- don't over crowd the pan. Cover the pan and cook until the bottom of the burgers have browned (Approx. 4 minutes). Turn the burgers, cover and cook until that side is brown as well. Use a meat thermometer and insert it in the thickest part of the burger and it registers 165 degrees.
Soak the wood chips in water for 30 minutes. Heat coals in a charcoal grill until they glow bright orange and ash over. Brush the grates with oil to prevent the burgers from sticking sticking. Place the burgers on the gill and grill the burgers until golden brown and slightly charred on the first side, about 5 minutes for turkey and 4 minutes for chicken. Flip over the burgers. Cook burgers until golden brown and slightly charred on the second side, until cooked- about 5 minutes on the second side. Use a meat thermometer and insert it in the thickest part of the burger and it registers 165 degrees.
Add the buttered rolls to the grill/pan and toast. Watch carefully not to burn. Remove the rolls and assemble burgers. Sandwich the hot burgers between the buns and serve immediately with desired toppings.