Serve with Martha's All-American Potato Salad.
Cut 2 slices bread into very small pieces, and place in a medium bowl. Add turkey, sour cream, 3 tablespoons chutney, 1 tablespoon mustard, chile powder, and salt; mix to combine.
Divide meat mixture into 8 equal-sized portions. Flatten one portion to a diameter of 3 1/2 inches. Place one quarter of the cheese in the center in a single layer. Top with another portion of meat and seal edges by pinching with fingers to enclose cheese. Repeat process with remaining ingredients. Place burgers on a parchment paper-lined baking sheet. Cover with plastic wrap, and freeze for 30 minutes.
Heat a grill or grill pan over medium high heat. Brush both sides of each burger with oil. Reduce heat to medium, and grill burgers until firm and cooked through, flipping once, 7 to 8 minutes per side.
Lightly toast remaining 8 slices of bread. Top each of 4 slices with 1/4 cup greens, followed by a burger and a tomato slice. Stir together remaining tablespoon each chutney and mustard. Spread chutney mixture evenly over remaining 4 toasted bread slices, and place on top to enclose. Serve immediately.