Place 2 halved and sliced bananas in a sealable quart glass jar and cover with 350 ml or 1-1/2 cups of vodka. Seal and shake.
Let the mixture steep for 3-4 days at room temp in a dark place. Strain out the fruit and filter everything through cheese cloth, pressing lightly to extract liquid. Repeat the filtering with fresh cheese cloth to remove any remaining solids
Heat 1/2 cup of water and sugar in a heavy bottom sauce pot on medium heat until it simmers, reduces and forms a syrup, about 10 minutes. Remove from heat and let it cool
Combine the syrup with the banana infused vodka. Seal in a bottle or jar, then shake to mix. Let it rest for one day. Store in the refrigerator until ready to use