In a blender, pulse blackberries until smooth. Using a fine sieve over a small bowl, strain the blackberry puree. Use a spoon to press the puree through the sieve. This will remove the larger pieces of pulp and seeds.

Add 2 tablespoons of sugar to the liquid blackberry puree and whisk to combine. Pour into a glass jar and set aside.

Pour the remaining sugar onto a small plate. Zest one lemon and add the zest to the sugar on the plate, mixing to combine. Slice the zested lemon into wedges and run a wedge around the rim of two margarita glasses (or regular glasses). Then, turn each glass upside down and dip the rim into the sugar, turning until evenly coated.

Fill each glass halfway with ice and set it aside.

Fill a cocktail shaker about three-quarters of the way with ice and add a tablespoon of blackberry puree, 1/4 cup lemonade, 2 ounces silver tequila, and juice from 1/2 of a lemon. Cover the cocktail shaker and shake vigorously for about 15 seconds. Place the cocktail strainer over the shaker and pour it into the glass.

Garnish with fresh blackberries and a lemon wedge on the rim of the glass. Repeat for the second margarita.