Preheat the oven to 425°F.

Place the peeled and thinly sliced peaches onto a rimmed baking sheet.

Sprinkle the brown sugar evenly over the peaches and toss gently to coat.

Roast in the hot oven until the peaches begin to break down, release their juices, and turn deeply golden, about 10 to 15 minutes.

Allow the roasted peaches to cool slightly.

Transfer the fruit and all pan juices to a blender and puree until completely smooth. (You should yield about 2 cups of thick peach puree).

In a medium saucepan, gently heat the half-and-half and vanilla extract over medium heat.

Heat just until tiny bubbles begin to form along the edges of the pan, about 5 to 6 minutes. (Do not let it boil).

Remove the pan from the heat and set aside.

In a separate, heavy-bottomed saucepan, combine the 1 cup of granulated sugar and ¼ cup of water.

Heat over medium until the sugar completely dissolves, about 2 minutes.

Increase the heat to medium-high and bring the sugar syrup to a rolling boil.

Continue boiling without stirring until the sugar begins to turn a deep caramel amber color, about 8 minutes.

Gently swirl the pan by the handle to ensure the sugar colors evenly.

Remove the caramel from the heat.

Very carefully and slowly pour the warm half-and-half mixture into the hot caramel. (Caution: The mixture will violently bubble up and expand).

If the bubbles rise too high, stop pouring, allow them to settle, and then continue pouring in the remaining liquid, whisking until smooth.

Place the 6 egg yolks into a medium mixing bowl and whisk them gently.

While whisking the eggs constantly, slowly drizzle in about ¼ cup of the hot caramel mixture. (This tempers the yolks so they do not scramble).

Pour the tempered egg yolks back into the main saucepan with the remaining caramel cream.

Reduce the stovetop heat to medium.

Stir the custard constantly with a wooden spoon until it thickens enough to heavily coat the back of the spoon, about 5 minutes.

Remove from the heat.

Pour the thickened custard through a fine-mesh sieve or colander into a clean, large mixing bowl to remove any cooked egg bits.

Stir the reserved roasted peach puree into the smooth custard base.

Create an ice bath by placing the custard bowl into a larger bowl filled with ice and cold water.

Stir occasionally until the mixture reaches room temperature, about 20 minutes.

Cover the custard tightly with plastic wrap pressed directly against the surface and chill in the refrigerator for at least 4 hours (or overnight).

Pour the deeply chilled custard into the bowl of your ice cream maker and freeze according to the manufacturer's directions.

Serve immediately for a soft-serve texture, or transfer to an airtight container and freeze until firm.