Use butter to grease the sides of a heavy 1-quart saucepan. In the saucepan, combine the sugar, corn syrup, water and vanilla. Cook over medium-high heat for about 5 minutes, to boiling, stirring constantly with a wooden spoon to dissolve the sugar. Avoid splashing the mixture on the sides of the pan. Carefully clip a candy thermometer to the side of the saucepan.
Cook the mixture over medium heat, stirring constantly, while adding the butter, 2 pieces at a time. Continue cooking the mixture over medium heat, stirring occasionally, until the thermometer registers 300° F., hard-crack stage. The candy mixture should boil at a moderate, steady rate over the entire surface. Reaching hard-crack stage should take 25 to 30 minutes.
Remove the saucepan from the heat; remove the thermometer. Stir in the vanilla extract. Cool for 5 minutes. Pour the mixture, 1 to 2 tablespoons at a time, onto the prepared baking sheets. The mixture will for 2 to 3-inch circles.
Quickly place a lollipop stick into each piece of candy, twisting gently to cover with the candy mixture.
Let the lollipops harden. Wrap the lollipops individually in clear plastic wrap to store at room temperature.
Makes 12 to 16 lollipops.