Combine 2 teaspoons sugar and warm water in a cocktail shaker; stir to dissolve sugar. Pour Meyer lemon juice, lemon peels, vodka, and orange liqueur into the shaker; add ice. Cover and shake vigorously.
Remove a lemon peel from the shaker and wipe the rim of a martini glass with the peel. Dip the rim of the glass in 1 teaspoon sugar, or as needed. Strain martini into prepared glass.