This is a basic recipe for duck burgers that you can modify as you need to suit your tastes. But what follows does make a great burger. Geese also work very well here, too.
2poundsground duck burger(see above for grinding instructions)
Salt and black pepper
2tablespoonsduck fat or vegetable oil
6burger buns
Instructions
Start by caramelizing the onions, which takes time. Sear the onions in the duck fat, tossing to coat, for a couple minutes, then lower the heat to medium-low and let them cook slowly, stirring occasionally, for 30 minutes. You may need to cover the pan to prevent the onions from sticking and burning. This step can be done a day or two in advance.
While the onions are cooking, made the aioli if you want. Put everything in the blender except for the melted duck fat. Buzz to combine, then, with the motor running at medium, drizzle in the melted duck fat. Puree until combined. It should look like a loose mayonnaise. Put this in the fridge to chill. This step can also be done up to 2 days in advance.
Turn your oven to 250°F and set a cooling rack over a baking sheet inside. This is for your burgers.
When you are ready to cook your burgers, make patties about 1/3 pound each. Heat a large pan over high heat, and when it's hot, add the duck fat or oil. When that fat is hot, salt your patties. Put one in the pan, then use a bacon press to squish the patty to the width of about 3/4 inch. Keep the press on the meat for 30 seconds, the slide it off.
Keep cooking the duck burger for another minute to 90 seconds, then flip. My advice is to use a metal spatula to scrape the patty off the pan, because it can stick a little and you want all the yummy brown bits on the patty.
Once flipped, top with some caramelized onions, and a slice of cheese, if you want. If you use cheese, cover the pan so it melts. Give the burgers between 1 and 3 minutes more cook time, then move the patty to the cooling rack in the oven. Repeat with the remaining patties. Most pans can do 2 burgers at a time.
If you want, when your patties are all done, toast your burger buns in the remaining fat in the pan, adding more if needed, until pretty and browned, which should only take maybe a minute. Build your burgers, top with the greens and the aioli, and serve.
Notes
Note that prep time does not include grinding the duck meat.
Keys to Success
Assuming you are grinding your own duck breast, grind it fine. This makes a better smash burger.
The duck fat aioli is great, but not 100% necessary. Make it if you have the ingredients, but if not, sriracha mayo or something similar is a good alternative.
Roasted, chopped green chiles are a fantastic alternative to the caramelized onions.
Eat these burgers quickly after making them because they are juicy and will soak the buns if you wait too long.