"He put on his hat with its peaks to the sides"

 Kutuzov is wearing a bicorne hat; they were very fashionable around the turn of the nineteenth century, and are now closely associated with the Napoleonic wars. The bicorne is descended from the tricorne, three-peaked hat, which was popular earlier in the eighteenth century.

The hat could be worn with its peaks to the sides ('athwart') or front-and-back ('fore-and-aft') depending on the mood of the wearer and the fashion of the time. The athwart position was fashionable in the 1790s whereas the fore-and-aft became popular in the 1800s; Tolstoy may be emphasising Kutuzov's age through his old-fashioned bicorne styling.