Purée turmeric and 1/4 cup water in a blender. Pour into a small saucepan and add 3/4 cup more water and the sugar. Bring to a boil, stirring. Remove from heat and steep 2 hours. Meanwhile, soak chiles in 1/3 cup warm water until softened, at least 10 minutes.
Strain turmeric syrup into a container and discard solids. Drain chiles, then purée with 1/4 cup lime juice in blender. Strain into syrup. Stir in remaining 3/4 cup lime juice and chill until cold, at least 1 hour.
Combine salt and chile powder on a small plate. Run a lime slice around rim of each glass, then dip rims in salt mixture. Making 1 cocktail at a time, shake 1 oz. each mezcal and tequila, 2 oz. turmeric syrup, and a handful of ice cubes in a cocktail shaker until shaker is cold. Strain into a chile salt-rimmed glass and float a lime slice on top.
*Find fresh turmeric at well- stocked grocery stores and Indian markets, or substitute 1 tbsp. ground turmeric. Find Indian red chile powder online and at Indian markets.