Place the coffee and 2 cups of the water in a jar or a pitcher, stir well. Cover and let sit at room temperature for at least 12 hours and up to 1 day. Strain through a coffee filter or strainer into a bowl working in batches.
Discard the grounds and transfer the coffee to a glass jar and set it aside.
To make the syrup, place the remaining ½ cup of water and sugar in a small pot over high heat and bring it to a simmer, stirring to dissolve the sugar. Remove from heat and let this syrup cool to room temperature.
Add the syrup, vanilla extract and aguardiente to the glass jar that contains the coffee and stir to combine. Tightly cover the jar and let it sit at room temperature for at least 2 weeks, shaking once per week.
Notes
This recipe can be double. Keep at room temperature up to 3 days or in the refrigerator for up to 1 month.