In an old-fashioned glass, swirl the absinthe to coat with a film, pouring out any excess. Fill the glass with ice.
In a cocktail shaker filled halfway with ice, gently stir the rye, simple syrup, Peychaud’s bitters, and Angostura bitters until ice-cold, 20–30 seconds.
Discard the ice in the glass. Strain the cocktail into the glass, then rub the rim with the lemon strip and drop it into the sazerac.