Butter a 9-inch pieplate. Set aside.
Place peanut butter and vanilla in a medium bowl.
Combine sugar and next 3 ingredients in a 2-quart heavy saucepan. Bring to a boil over medium heat, stirring until butter melts. Using a pastry brush dipped in hot water, brush down any sugar crystals on sides of pan. Cook, without stirring, until a candy thermometer registers 235° (soft ball stage). Remove from heat, and let stand 5 minutes. Pour hot syrup over peanut butter mixture in bowl. (Do not scrape sides of pan.) Beat with a wooden spoon until smooth. Quickly pour into prepared pieplate. Cool completely. Cut candy into pieces.
Tips: Grandma probably used the stage method to determine her candy's readiness, but a thermometer will give you more consistent results. Be sure to check that your thermometer works accurately and adjust accordingly if it doesn't. For example: if water boils at 212° on your thermometer, your thermometer works accurately. If it boils at 210°, cook this recipe to 233°; if the boiling point on your thermometer is 214°, cook this recipe to 237°. Be sure that the bulb of the candy thermometer is fully submerged in syrup but does not touch bottom of saucepan. Don't beat candy too long; it hardens quickly as it cools. Beating candy in a separate bowl helps prevent the grainy texture that results from accidentally scraping any crystals off sides of pan.