Cut kernels from corn cobs; reserve cobs. Break each cob into 2-3 pieces. Bring milk to a simmer in a large saucepan. Add corn kernels and cobs. Remove mixture from heat, cover, and let steep for 45 minutes.
Remove cobs from milk; discard. Purée mixture in batches in a blender.
Set a coarse strainer over a large bowl. Strain mixture, pressing on solids; discard solids.
Add more milk if needed to measure 3 1/2 cups.
Bring corn mixture, 1 1/4 cups sugar, and cream to a simmer in a large saucepan over medium heat, stirring to dissolve sugar.
Set a strainer over a medium bowl; set aside. Whisk remaining 1/4 cup sugar, egg yolks, and salt in a medium heat proof bowl. Gradually whisk in hot milk mixture; return to saucepan. Stir constantly over medium heat until custard registers 175° on an instant-read thermometer, about 2 minutes.
Immediately pour custard through strainer. Place bowl with custard over a large bowl of ice water. Let stand until cold, stirring occasionally, about 5 minutes. Cover and refrigerate custard for at least 6 hours or, preferably, overnight.
Process custard in an ice cream maker according to manufacturer's instructions. Transfer to a container; freeze for at least 1 hour and up to 1 day.