In a medium, heavy-bottomed saucepan over medium heat, melt 3/4 cup of the sugar, stirring often, until it takes on a nice sandy color and is a smooth liquid, 10 to 15 minutes.
Quickly add the fruit and stir to coat with the sugar; this will prevent the sugar from caramelizing any further and/or burning. Cook until the fruit is soft, about 10 minutes or so, depending on which fruit you're using and its ripeness.
Transfer the caramelized fruit to a blender. Add the miso and process to a smooth purée. Set aside.
To make the ice cream custard, heat the milk and heavy cream in a saucepan.
While the milk is warming, make an ice bath by placing a 2-quart (2l) bowl in a larger bowl partially filled with ice and water. Set a strainer over the top of the smaller bowl and pour the cream into the bowl.
In a separate bowl, whisk together the egg yolks and sugar. Gradually pour some of the warm milk into the yolks, whisking constantly as you pour. Scrape the warmed yolks with milk back into the saucepan.
Cook over low heat, stirring constantly and scraping the bottom with a heat-resistant spatula, until the custard thickens enough to coat the spatula. Strain the custard into the heavy cream. Stir over the ice until cool. Add the fruit-miso mixture and stir to mix completely. Add the vanilla extract and refrigerate to chill thoroughly, preferably overnight.
Churn in your ice cream maker according to the manufacturer's instructions.