Combine first 3 ingredients in a medium saucepan, and cookover medium-high heat until sugars dissolve. Bring to a boil; reduce heat, and simmer 5 minutes. Place pecans and vanilla beans in a1-gallon jar. Pour hot mixture into jar; let cool. Add vodka; stirwell. Cover tightly, and store in a dark place at room temperature at least 2 weeks. Shake jar gently once daily.
Pour mixture through a wire-mesh strainer lined with 2 layers of cheesecloth into a bowl, discarding solids. Pour mixture through a wire-mesh strainer lined with a coffee filter into a bowl. Change filter often. (Mixture will drip slowly.) Pour mixture into jars; cover tightly. Store at room temperature.
Use in coffee, baked apples, or low-fat cakes.