In a medium mixing bowl, stir together the yogurt and lemon zest and juice. Season, to taste, with a couple pinches of salt. Sauce can be made ahead of time and refrigerated until ready to use. Bring to room temperature before using.
Cut the brioche into about 1/3 to 1/2-inch thick slices and then use a small glass to cut out as many circles as possible, about 3 inches in diameter, from each slice. Place the bread "buns" in an airtight container until ready to use so that they stay moist and soft.
Make the burgers:
Preheat the grill.
Combine all ingredients in a mixing bowl and work together until they are fully incorporated. Using a small ice cream scoop, scoop the mixture into little balls about 1 to 1 1/2 inches big and place them on a parchment paper-lined baking pan. Using your hands, flatten burgers to about 1/2-inch thick and just a couple inches in diameter.
Once the grill is hot, place them on the grill and cook, about 2 to 3 minutes per side or until fully cooked through; cook time varies based on how hot your grill is.
Place the cooked mini burgers on top of the cut brioche and add a piece of lettuce and a tomato slice. Top with a dollop of the lemon yogurt sauce and serve.