Heat half-and-half to 175°F in a large, heavy saucepan. Stir in the salt and 1 ¼ cups of the sugar and whisk until dissolved.
Ladle a small amount of the half-and-half mixture into the eggs. Pour egg mixture into the pan, whisking constantly. Continue to cook over low heat, stirring constantly, until mixture reaches 160°F and coats the back of a metal spoon.
Remove from heat. Place pan in a bowl of ice water; stir for 2 minutes. (Alternately, you can fill your sink with an inch of cold water & ice cubes and use that instead of a bowl.) Stir in whipping cream, 2.5 oz of the evaporated milk, and vanilla. Press plastic wrap onto surface of custard and refrigerate for several hours or overnight.
After making the custard and at least 3 hours before starting the ice cream maker, make the fudge ribbon; heat remaining evaporated milk and sugar over low heat, whisking constantly. When mixture is heated, add chopped chocolate and heat until melted, whisking constantly. Remove from heat and allow mixture to come to room temperature.
Remove custard from refrigerator and stir in the banana puree. Fill cylinder of ice cream maker with the banana custard mixture and freeze according to manufacturer's directions. (Depending on the size of your ice cream maker, you might need to make two batches.)
When ice cream is frozen, transfer it to an airtight freezer container. Fold in fudge ribbon, peanut butter, and peanut butter cup pieces, careful not to overmix. Freeze for 2–4 hours before serving.