Here's how I do duck, and it turns out great. I'd imagine goose would be similar, though I haven't tried one yet.
Make enough brine to fully cover the bird (I usually make a two gallon batch for 2 or 3 ducks but it depends on the size of the birds and the pot you're putting them in)
For each gallon of water add the following:
- 3/4 cup non-iodonized salt (pickling salt)
1 cup brown sugar
3 cups apple cider
1/2 cup lemon juice
1/4 cup Worcestershire sauce
1 oz maple flavoring
1 tsp ginger
3 Tbsp ground black pepper
2 Tbsp minced garlic
1 cup dry minced onions
2 bay leaves
2 Tbsp Italian seasoning
put the brine in a stainless-steel or porcelain container, add the bird(s) and soak overnight in the fridge.
Next day, take the bird(s) out of the brine and bring the brine to a slow boil. Boil the bird(s) about 3 or 4 minutes per pound (I usually never go over 15 minutes though) this step kills any bacteria on the outside or in the cavity of the bird, it's not really to cook it.
While the brine is getting up to boil, start the smoker and get it up to around 225 degrees.
Put the bird(s) in the smoker and sit back for a few beers, this step could take anywhere between 4 and 16 hours depending on how hot your smoker gets, the weather and how big the birds are. Your aim is to get the internal temperature of the bird up to 160 degrees or so.
If you're short on time, you can smoke the birds in the smoker for 3 or 4 hours and then finish it up in the oven.