Combine evaporated milk and pudding mix in a saucepan over medium-low heat. Stir and simmer until thickened, about 10 minutes.
Remove saucepan from heat. Stir in sugar, eggs, and vanilla extract and pour mixture into an ice cream maker. Pour milk to the fill line and stir. Freeze according to manufacturer's instructions, about 20 minutes. Transfer to an airtight container and freeze until firm, about 4 hours.