For the vinaigrette: In a small bowl, mix together the first five ingredients. Slowly whisk in the olive oil to emulsify, then chill.
For the tzatziki: Whisk together all ingredients and chill.
For the Burgers: Place the lamb and the rest of the ingredients in a large bowl and mix together. (Handle the lamb as little as possible to avoid compacting it.)
Flatten into patties.
Using your thumbs, make a deep well in the center of each patty, fill each with one quarter of the crumbled feta and pull some ground lamb up to cover.
Flatten again, wrap the patties in plastic wrap and chill for at least an hour before cooking.
To cook, sear the patties in a cast-iron skillet or on the grill for about 3 to 4 minutes per side (for pink in the center), until slightly charred, flipping once.
Note: patties also freeze well after forming for thawing and cooking later.
Serve over field greens, baby spinach, thinly sliced red onions, cucumbers, cherry tomatoes, red pepper rings, Kalamata olives, pita crisp croutons, additional crumbled feta, roasted walnuts or pistachios, fresh dill and mint sprigs and/or nasturtiums, and pass the dressing and tzatziki at the table.