Melt the unsalted butter in a heavy-bottomed saucepan over medium heat. Continue cooking, stirring occasionally, until the butter foams, turns golden brown, and nutty-smelling flecks appear at the bottom, about 5-7 minutes. Be careful not to burn it!

Add the diced peaches, 1/4 cup of the granulated sugar, and lemon juice to the browned butter. Cook, stirring occasionally, until the peaches soften and release their juices, about 8-10 minutes. Remove from heat and let cool completely. A pinch of the divided salt can be added here.

In a separate medium saucepan, combine the heavy cream, whole milk, the remaining 1/4 cup granulated sugar, and the remaining salt. Heat over medium-low heat until the mixture is steaming and small bubbles form around the edges, but do not boil. In a large bowl, whisk the egg yolks until smooth. Slowly ladle about half of the hot cream mixture into the egg yolks, whisking constantly to temper them. Pour the tempered yolk mixture back into the saucepan with the remaining cream mixture. Cook over medium-low heat, stirring constantly with a spatula, until the custard thickens enough to coat the back of a spoon (it should reach about 175°F-180°F), about 5-8 minutes. Do not boil.

Remove the custard from the heat and stir in the vanilla extract. Pour the custard through a fine-mesh sieve into a clean bowl to remove any cooked egg bits. Stir in the cooled brown butter peach mixture. Cover the bowl with plastic wrap, pressing it directly onto the surface of the custard to prevent a skin from forming. Chill the mixture completely in the refrigerator for at least 4 hours, or preferably overnight.

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

Transfer the churned ice cream to an airtight freezer-safe container. Freeze for at least 4 hours, or until firm, before serving.