How do you get the warm doublet?

There are 3 ways to get it from the Old Man:

• Read the diary in his hut on the Great Plateau, then cook a meal using a spicy pepper, meat and a Hyrule bass. Show him the meal, and he will give you the warm doublet.
• Climb to the top of Mount Hylia on the Great Plateau (before you complete the trials on the Great Plateau). He will appear and give you the warm doublet as a reward.
• After completing the Great Plateau, he will leave a warm doublet in his hut.

Once you obtain it from one method, you can't obtain it from the other 2. But you can still buy it from Hateno Village later in the game for 80 rupees.

You don't have to have the warm doublet to travel through the cold areas. Hint: try cooking.

The Spicy Meat and Seafood Fry dish will give you low-level cold resistance.

A number of sites, including shacknews and gamesradar, have guides on this topic. It's also possible to just buy the Warm Doublet, after you leave The Great Plateau.