How to get the warm doublet
If you want the warm doublet before you head to the snowy area to tackle the two shrines found there, you need to make a Spicy Meat and Seafood Fry for the Old Man. If you complete the shrines without it, you'll find the doublet in the Old Man's cabin, but it's much more convenient to pick it up beforehand.
Read the diary in the Old Man's cabin and you'll discover a partial recipe. It turns out that he'll hand his warm doublet over to whoever figures out the missing ingredient.
The three ingredients are a Spicy Pepper, Raw Meat, and Hyrule Bass.
The pepper can be found growing near the entrance to the snowy area at the end of the path from the Shrine of Resurrection. There is a Bokogoblin camp off to the right of the arch and you'll find a pepper plant here.
The fish can be seen swimming in the lake next to the Temple of Time. You can either shoot them with an arrow and dive in to grab them, swim over them and grab them, or throw a bomb in the water and wade in to get their bodies.
The Forest of Spirits is home to a number of boar. The easiest way to kill them is with a well-placed headshot. Otherwise, you can try to sneak up on them, but be warned, they're flighty buggers.
Once you've made the dish, you need to present it to him when he's resting next to a campfire at night and tell him you cooked something.
Hand over the dish and the warm doublet is yours.