Combine the quartered figs, 0.25 cup granulated sugar, balsamic vinegar (if using), and water in a medium saucepan. Bring to a gentle simmer over medium heat and cook for 8-10 minutes, stirring occasionally, until the figs are very soft and the mixture is slightly syrupy. Remove from heat, and with the back of a spoon or a fork, gently mash about half of the figs to create a rustic compote, leaving some chunks for texture. Let cool completely to room temperature, then chill in the refrigerator for at least 30 minutes.

In a large heatproof bowl, whisk the 0.75 cup granulated sugar and egg yolks together until light in color and thick. Set aside.

In a clean medium saucepan, combine the whole milk and heavy cream. Heat over medium heat until it just begins to simmer around the edges, but do not let it boil. Remove from heat.

Slowly pour about half of the hot milk mixture into the egg yolk mixture, whisking constantly to temper the yolks and prevent scrambling. Pour the tempered yolk mixture back into the saucepan with the remaining milk. Return the saucepan to low-medium heat and cook, stirring constantly with a rubber spatula, for 5-8 minutes, until the custard thickens enough to coat the back of the spatula. Do not let it boil.

Remove the custard from the heat and immediately stir in the drained ricotta cheese, vanilla extract, and pinch of salt until well combined and smooth. Pour the mixture through a fine-mesh sieve into a clean bowl to ensure a super smooth texture. Cover the bowl with plastic wrap, pressing it directly onto the surface of the custard to prevent a skin from forming. Chill in the refrigerator for at least 4 hours, or preferably overnight, until completely cold.

Once the base is thoroughly chilled, pour it into your ice cream maker and churn according to the manufacturer's instructions. This usually takes 20-30 minutes, until it reaches the consistency of soft-serve ice cream.

Transfer about half of the churned ice cream to a freezer-safe container. Spoon half of the chilled fig compote over the top. Add the remaining ice cream, then dollop with the remaining fig compote. Gently swirl with a knife or spoon to distribute the fig mixture, but don't overmix. Cover tightly and freeze for at least 4-6 hours, or until firm, before serving.