Make the candied bacon: Preheat the oven to 350 degrees F. Mix the brown sugar, cayenne and cinnamon in a small bowl. Arrange the bacon on a wire rack set over a rimmed baking sheet. Sprinkle with half of the sugar-spice mixture. Bake 10 minutes, then flip and sprinkle with the remaining sugar-spice mixture and bake 10 more minutes. Remove the bacon from the oven and let cool.
Make the bacon mayonnaise: Cook the bacon in a skillet over medium heat, turning occasionally, until crisp, 9 to 11 minutes. Drain on paper towels, then finely dice. Whisk the mayonnaise and sour cream in a small bowl. Stir in the diced bacon and chives and season with salt and black pepper. Cover and refrigerate until ready to use, or up to 1 day.
Make the burgers: Form the ground beef into 4 equal-size patties, about 3 1/2 inches in diameter. Sprinkle both sides of each patty liberally with salt and black pepper.
Heat a grill pan or cast-iron skillet over high heat, then add the canola oil. Add the patties and cook about 3 minutes, then flip and top each with 1 slice cheddar. Cover and continue cooking 3 more minutes for medium rare.
Build the burgers: Smear the bacon mayonnaise on the toast. Serve the patties on the toast with lettuce, tomato and the candied bacon.