If not already done, clean fish and remove skin. Cut into pieces, approximately 2 inch squares.
Place the fish in a pot, add the onion and salt, and cover with the water.
Bring just to the boiling point and reduce to a low simmer. Simmer until the fish flakes when poked with a fork. Check often, you don't want it to get mushy.
Drain the pot and set aside the fish & onions. The water is now the fish stock for the soup; return it to the pot and add the potatoes.
Simmer the potatoes in the stock until tender. Time will depend on the size of the pieces, up to 15 minutes or so.
Return the salmon and onion mixture to the soup and add the chives and the milk.
Simmer at a very low simmer for 20 minutes.
If you are not going to serve the soup in the pot you prepared it in, transfer it before you add the butter.
Slice the butter into thin pats and place them on the top of the soup.