Make simple syrup, combine 1 cup water with 1 cup sugar in a small saucepan and heat over medium heat, stirring constantly, until sugar is dissolved. Cool before using.
Place the blackberries and lime zest in a sealable glass jar, muddle lightly to release juice,
add brandy and vodka. Seal and shake. Let mixture steep for 3 days at room temperature away from direct sunlight.
Strain through a fine mesh strainer, pressing down to extract juice, then filter mixture through a coffee filter or through two layers of cheesecloth. Discard solids.
Combine blackberry infusion and simple syrup in a sealable bottle or jar, then shake to mix. Let rest for a minimum of one day. Store in the refrigerator for up to two months.
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