Both make the drink colder by removing warmth.
Stirring removes warmth without adding air for a silky texture. This works for recipes that are just alcohol based like Manhattans because warm alcohol tastes and feels hot.
We shake drinks with citrus/cream/eggs because they either feel sharp and we need to make them smooth, or they feel heavy and we need to make them feel light. Shaking hard makes cold and adds fluff. We like cold and fluff in drinks.
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