Divide vodka, peach, fresh thyme and lemon zest between 2 sealable glass quart jars. Seal and shake. Let mixture steep for 2-3 days at room temperature away from direct sunlight.
Sample before straining to see if the peach flavor is sufficient. If not, let steep for another day.
Strain fruit, then filter mixture through a coffee filter or through two layers of cheesecloth, pressing down to extract liquid.
Heat water and sugar in a pot on medium heat until it boils and forms a syrup, about 5 minutes. Let syrup cool.
Once the syrup is cool, combine it with the peach-zest infusion. Seal in bottle or jar, then shake to mix.
Let rest for a minimum of one day.
Store in the refrigerator for up to two months.