In a bowl, mix together all the ingredients for the yogurt and keep covered in the fridge until ready to serve. This mint yogurt is even better if you let the flavors sit together for a few hours before serving.
In a large bowl, mix together all of the ingredients for the burgers, making sure everything is evenly incorporated, but being careful not to over mix. Form 4 equally sized patties. You can do this in advance and keep the patties in the fridge until ready to grill.
With the grill on medium low flame, add the burgers and cook, flipping once, until they are medium to well done, depending on your preference. Alternatively, you can sear the patties in an oven proof skillet coated with oil, flipping once, until nicely browned on both sides and finish in a 350°F oven, about 7-10 minutes longer.
While the burgers are cooking, cut 4 thick slices from the fresh country loaf. If the loaf is large, you can cut 2 slices and halve them so the piece of bread is just larger than the burger. Brush the bread with olive oil and rub with the garlic clove. Toast on the grill or pop into a 350°F oven until toasted while the burgers are cooking.
Once everything is cooked, place the toasted bread slices on the plate and top with the cooked burger. Put a generous dollop of the yogurt on the burger and then a scattering of the lettuce. Serve immediately.