Place 14 blackberries and 2 thyme sprigs in medium bowl. Press firmly on solids with muddler or back of wooden spoon until mashed. Mix in tequila, lime juice, Simple Syrup, and Cointreau, then 1 cup ice. Stir to blend well.
Strain into large glass measuring cup. Mix in sparkling wine. Divide remaining ice between 2 tall glasses. Pour Margarita mixture over. Garnish with 1 blackberry each and 1 thyme sprig.
Stir sugar and water in medium saucepan over medium heat until sugar dissolves. Increase heat and bring to boil. Reduce heat and simmer 3 minutes. Cool. (can be made 2 weeks ahead. Refrigerate in airtight container).