Combine all of the ingredients for the burgers except for the salt and oil in a small bowl and mix gently until combined.
1 pound ground duck, 1 egg, ¼ cup cilantro, chopped, 1 clove garlic, minced, 1 green onion, green parts only, chopped fine, ¼ teaspoon black pepper
Divide the mixture into fourths, shape the duck burgers, and leave them on wax paper at room temperature for 20 to 30 minutes.
Heat a cast iron skillet or griddle and add the oil.
1 tablespoon vegetable or other neutral oil for frying
When the oil is hot, sprinkle half of the salt onto the patties and put them in the skillet salt side down.
½ teaspoon salt
Use your thumb to create a concave divot in the center of each burger and cook for 3 to 4 minutes.
Sprinkle the remaining salt on the up side of the burgers and flip them over.
Add cheese if melted cheese is desired. Cook for another few minutes until the internal temperature is at least 165°F.