Combine sugar, milk, egg, and butter in a medium-size heavy saucepan. Cook over medium heat, stirring constantly, until sugar dissolves. Cover and continue to cook 2 to 3 minutes to wash down sugar crystals from sides of pan. Uncover and cook, stirring frequently 20 minutes or until mixture reaches soft ball stage (234°).
Remove from heat. Cool to lukewarm (110°). Stir in vanilla and pecans; beat until mixture thickens slightly. Pour into a buttered 8-inch square baking pan. Cool slightly; cut fudge into 1 1/2-inch squares. Store in airtight containers.