Step 1

Fill a large bowl halfway with ice water; set aside. Heat half-and-half in a medium saucepan over medium-high, stirring occasionally, until a thermometer registers 175°F, about 5 minutes. Whisk in sugar and salt until dissolved. Place eggs in a medium bowl; gradually whisk in 1/4 cup of the half-and-half mixture. Pour egg mixture into remaining half-and-half mixture in saucepan. Reduce heat to medium-low; cook, stirring occasionally, until thermometer registers 160°F, 5 to 8 minutes. Remove from heat.

Step 2

Holding the saucepan handle with 1 hand, submerge bottom of saucepan into bowl of ice water; stir constantly for 2 minutes. Pour cooled mixture in saucepan into a medium bowl. Stir in whipping cream, evaporated milk, and vanilla. Place plastic wrap directly on surface; chill at least 30 minutes or up to overnight (or 8 hours).

Step 3

Add mashed bananas to chilled cream mixture; stir until smooth. Pour mixture into frozen freezer bowl of a 2 1/2- to 3-quart electric ice-cream maker, and proceed according to manufacturer's instructions. (Instructions and times may vary.) Transfer to an airtight freezer-safe container; freeze until firm, about 2 hours. Serve with desired toppings.