Banana-Caramel Ice Cream

INGREDIENTS

• 4 large egg yolks
• 1 2/3 cups sugar, divided
• 1 2/3 cups whole milk
• 1 2/3 cups plus 1/2 cup heavy cream, divided
• 3 to 4 very overripe bananas (skin should be more brown than yellow), mashed (about 1 1/2 cups)
• 1 teaspoon pure vanilla extract
• 1 teaspoon salt, divided
• 1/4 cup water
• 1 tablespoon light corn syrup
• 1/2 stick cold unsalted butter, cut into small pieces