The secret to the intense banana flavor of this banana-caramel ice creamis starting with very overripe fruit.
Vigorously whisk together egg yolks and 2/3 cup sugar in a medium bowl until pale yellow and slightly fluffy, about 2 minutes. Meanwhile, bring milk and 2/3 cup cream to a boil in a medium saucepan. Remove from heat, and gradually whisk into egg mixture. Pour mixture into pan, and cook over medium-high heat, stirring constantly with a heatproof spatula, 1 minute. Reduce heat to medium, and cook until thick enough to coat the spatula, about 1 minute (do not let boil). Pour custard into a bowl set in a larger bowl of ice water, and let cool, stirring occasionally. Puree bananas with half the custard in a blender until smooth, and transfer to bowl with remaining custard. Stir in vanilla, 1/4 teaspoon salt, and 1 cup cream until combined, and pour through a fine sieve into another bowl. Refrigerate until cold, at least 2 hours but preferably overnight.
Place water, corn syrup, and remaining sugar in a medium saucepan. Bring to a simmer over medium heat, swirling pan occasionally (do not stir). Raise heat to high, and continue to cook in same manner until mixture turns deep amber, about 9 minutes more. Remove from heat, and slowly pour in remaining cream. Add remaining salt, and simmer caramel 1 minute. Place pan in a bowl of ice water, and let cool until warm but not hot. Remove from ice bath, and whisk in butter, a few pieces at a time, until fully incorporated.
Place a 5-by-10-inch loaf pan (9 cups) in freezer at least 10 minutes. Freeze custard in an ice cream maker according to manufacturer's instructions. Spread one-third of ice cream in pan, and drizzle with one-third of caramel (it should be fluid but just barely warm; reheat slightly if necessary). Swirl caramel into ice cream with a wooden skewer. Repeat 2 more times until all the ice cream and caramel have been used. Freeze until firm, at least 4 hours and up to 1 day.