Cut each of 3 vanilla beans into 4 pieces; split each piece lengthwise. Place in a 1 1/2 quart bottle and add vodka. Cover tightly and shake vigorously. Let stand in a cool dry place 3 weeks, shaking bottle every 2 days.
Line a funnel with a coffee filter; pour mixture through a funnel into a bowl, discard beans.
Cook sugar and water in a small saucepan over medium-high heat, stirring occasionally, 2 to 3 minutes or until sugar dissolves. Remove from heat; cool completely. Add to vodka mixture.
Cut remaining 3 vanilla beans into pieces. Fill small bottles with 1 vanilla bean piece and extract; cover tightly and let stand in a cool dry place 1 month. Strain vanilla, discarding beans. Pour into decorative bottles.