1. In heavy 4-quart saucepan, mix sugar, flour, and salt; stir in milk. Cook over medium heat until mixture thickens slightly and boils, stirring frequently.
2. In medium-sized bowl, beat eggs slightly; stir in small amount of hot mixture. Slowly pour egg mixture back into hot mixture, stirring rapidly to prevent lumping. Cook over low heat, stirring constantly, until mixture thickens, about 5 minutes (do not boil or mixture will curdle). Remove saucepan from heat. Cover surface of custard mixture with plastic wrap; refrigerate, stirring occasionally, until completely chilled, about 3 hours.
3. Pour chilled custard mixture, heavy cream, and vanilla extract into 5- to 6-quart ice-cream freezer can. Place dasher in can and cover. Place can in freezer bucket; attach motor or hand crank. Add water to bucket.
4. Fill bucket half full with ice; sprinkle with 1/4 cup rock salt or 3 tablespoons table salt. Add about an inch of ice and 1/4 cup rock salt or 3 tablespoons table salt. Repeat layers of ice and salt until layers reach 1 inch below can lid. Freeze as manufacturer directs, adding more ice and salt as needed. Freezing takes about 35 minutes.
5. When freezing is finished, ice cream will be soft. Remove motor or crank; wipe lid clean. Remove dasher. With spoon, pack down ice cream. Cover surface of ice cream closely with plastic wrap. Replace lid and put cork in hole in center; add more ice and salt to completely cover lid. Let ice cream stand to harden, about 2 to 3 hours, adding ice as needed. (Or place container in home freezer for 2 to 3 hours.)