Add ice to one of your mixing tins and set it aside while you gather the remainder of your ingredients. This will give the ice time to temper, or melt slightly to slowly raise its surface temperature to avoid the ice cracking when the warmer liquid is added.
Take out the glass you intend to serve the cocktail in. A coupe or martini glass is recommended for a Manhattan served neat, while a rocks/low-ball glass is recommended for a Manhattan served on the rocks.
Measure and pour the rye into your second mixing tin.
Measure and pour the sweet vermouth into your second mixing tin with your rye. Note that short measures, like the "short oz" used to measure the sweet vermouth, denote that you should measure slightly less than stated amount of the ingredient. Being exact isn’t as important as being consistent, so when first starting out it may be easier to pour out a barspoonful (1 tsp) of a measured 1 oz of the ingredient to make a short oz.
Add your bitters to the rye/sweet vermouth mixture. (Hint: when you first turn the bottle upside down into your tin the liquid that comes out counts as your first dash)
Cover your first mixing tin with your strainer, turn it upside-down over the sink and give it a little shake. This allows any excess water that melted as the ice tempered to drain out, helping to avoid diluting your cocktail.
Pour the mixture into the tin with the ice, or (if you're looking to avoid having to wash the extra tin) carefully pour the ice into the tin with the mixture, being careful not to splash.
Stir the mixture until chilled. You should see condensation form outside the tin when the mixture is chilled. With practice, you will be able to consistently stir for a set number of seconds instead of looking for condensation.
Strain the mixture into your glass.
Add your garnish of choice to the cocktail.
Enjoy!