To make the thyme simple syrup, combine thyme, sugar, and water in a small saucepan over medium-high heat. Bring the mixture to a boil and stir to dissolve sugar. Remove from heat and cover. Steep for 30 minutes. Remove thyme sprigs and let cool completely.
To make the blackberry puree, combine blackberries, brandy, lemon juice and black food coloring in a blender or food processor. Puree until smooth. Set in the refrigerator until you make the cocktail.
In a cocktail shaker filled with ice, add Absolute Elyx vodka, blackberry puree and thyme simple syrup. Shake well and strain into a martini glasses.
Top (carefully!) with a small spoonful of crushed dry ice for the smoky effect!
Note: use caution when consuming the beverage with dry ice in it. I generally let it smoke and get cold, before removing dry ice chunks that didn't melt away at the bottom of the glass.