Start by mixing the thyme leaves and salt. Massage this mixture into the duck or goose legs, making sure everything is coated well. If there is any left over, pour it into a non-reactive container.
Place the salted duck or goose legs in the container, cover and refrigerate for 12-24 hours. The longer you go, the saltier your rillettes will be. I like to salt them down in the evening and cook them around mid-day the following day.
When you are ready, rinse off the cure well. Arrange the legs in a large pot and cover with the duck stock; add water or white wine if the level of the liquid does not cover the legs. Add the bay leaves.
Cover, then either cook on the stove over low heat or in a 225 degree oven for 4-8 hours. You want the meat to be falling off the bone. Check after about 2 hours to see if you have enough liquid. Uncover the pot in the final hour or two to evaporate the liquid a bit.
When the meat is falling off the bone, turn off the heat and let the duck or goose legs cool. Ideally, you refrigerate everything to solidify any fat (you can use it later).
When it is cool, pull off the skin and discard, then shred the meat and place in a large bowl.
Taste the remaining stock to see how salty it is. You will need a little for the rillettes, but you want to know how much to add — without oversalting the rillettes.
Add a little of the stock to the rillettes and beat together with a stout wooden spoon until the liquid is incorporated.
Add 2 tablespoons of the duck fat and beat it in. Keep adding duck fat until the mixture becomes creamy. Do this little by little.
When a spoonful of the rillettes tastes creamy, not dry-ish, add 2 tablespoons Armagnac and half the rue. Beat it in well. Taste it: You should be able to taste both the Armagnac and the rue, but neither should be overpowering. If you want, add the remainder of the liquor and herbs and beat them in well.
Press the rillettes into ramekins, or just a non-reactive container that works. Tupperware is OK, and a butter crock is ideal. Top with about 1/4 inch of duck fat. Cover with a lid or foil and store in the fridge. It is best to let this ripen for a week before serving, but it is perfectly fine eaten straight away. It’ll keep for at least two months, and if there are no air pockets and it is covered in fat, the rillettes will last up to 6 months.