anneal, temper, burn
焠 centers on applying intense heat to alter material properties: annealing or tempering metal, and more broadly burning or heating something strongly.
焠 combines 火 (fire) with 卒, which likely contributes the sound and may relate to suddenness or completion, suggesting the rapid heating and cooling of annealing.
Fire 火 on the left and 卒 on the right: imagine a soldier (卒) holding a blade in the fire to anneal and temper it, making it stronger.
For サイ, picture a sigh of relief as the hot metal is cooled: sigh -> サイ, and the fire 火 tempers the blade.