Make a simple syrup by stirring together and bringing 2/3 cup sugar and 1/3 cup water to a boil until the sugar dissolves; set aside to cool.
Make the sour mix by briskly stirring together the 2 1/4 cups water, lemon juice, 1/4 cup sugar, and egg white; set aside.
Combine the ice cubes, tequila, Cointreau, lime juice, 1 ounce of the simple syrup, and 1/2 cup of the sour mix in a pitcher; stir vigorously to infuse the ice, 8 to 15 seconds.
Run a lime wedge along the rim of an 8-ounce glass. Spread the kosher salt onto a plate and dip the glass rims in the salt to coat; add ice if desired. Strain the mixture from the cocktail shaker into the glass to serve.