Freeze 1/4 cup raspberries for at least 30 minutes prior to making the cocktail.
In an empty shaker, place 6-8 raspberries. Muddle the raspberries.
Add the pomegranate juice, mint simple syrup, vodka, and lime juice. Add ice and shake vigorously.
Place 4-5 raspberries in the bottom of a martini glass (as a garnish) and pour the contents of the shaker into the martini glass.
Place 4-5 frozen raspberries into a chilled martini glass and pour the contents of the shaker over the raspberries.
In a small saucepan, combine the sugar, water, and mint and bring to a boil. Allow mixture to boil for 2-3 minutes, then reduce heat to a simmer and simmer for 5 minutes. Allow the mixture to cool, then remove the mint leaves.
Store in a sealed jar for up to 1 week in the refrigerator.