In a large saucepan, heat 2-1/2 cups milk to 175°; stir in the sugar until dissolved. Whisk a small amount of hot mixture into the eggs. Return all to the pan, whisking constantly. Cook and stir over low heat until mixture reaches at least 160° and coats the back of a metal spoon. Remove from the heat. Cool quickly by placing pan in a bowl of ice water; stir for 2 minutes. Stir in vanilla.
Place remaining milk in a bowl; whisk in pudding mixes for 2 minutes. Let stand for 2 minutes or until soft-set. Stir into egg mixture. Stir in whipped topping. Press waxed paper onto surface of custard. Refrigerate for several hours or overnight.
Fill cylinder of ice cream freezer two-thirds full; freeze according to the manufacturer’s directions. Refrigerate remaining mixture until ready to freeze. Transfer to a freezer container; freeze for 2-4 hours before serving. Yield: 2-1/2 quarts.