In a medium mixing bowl, stir together the yogurt, lemon zest and juice and 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 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 in an airtight container until ready to use so they stay moist and soft.
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 of inches in diameter. Once the grill is hot, place them on the grill and cook, about 2-3 minutes per side or until fully cooked through, cook time varies based upon 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.