Whisk egg yolks and 1/4 cup sugar in a large bowl. Stir cream and remaining sugar together in a saucepan over low heat; bring to a simmer, about 12 minutes. Remove from heat. Slowly pour cream mixture into egg yolk mixture, whisking constantly.
Return mixture to the saucepan; cook over low heat, whisking constantly, until thickened and bubbly. Strain mixture through a fine-mesh sieve into a bowl, then whisk in buttermilk, vanilla, and salt. Let cool completely, at least 30 minutes. Chill at least 4 hours or up to overnight.
Pour mixture into an ice cream maker and freeze according to manufacturer’s instructions, about 20 minutes. Transfer to an airtight container and freeze until firm, about 4 hours.
Preheat the oven to 350 degrees F (175 degrees C) for the cobbler.
Wash the fuzz off of the peaches and cut into wedges. Toss peach wedges with 1/2 cup sugar.
Place butter in the bottom of a 9×13-inch casserole dish and melt it in the oven, 5 to 8 minutes.
Meanwhile, mix together flour, remaining 1/2 cup sugar, salt, and cinnamon. Stir in half-and-half, vanilla, and almond extract to form a loose batter; pour into baking dish over melted butter. Do not stir. Spoon peaches evenly over batter, without stirring.
Bake in the preheated oven until a toothpick inserted into the center comes out clean and peaches are tender, 45 to 50 minutes. Serve warm with ice cream.
Substitute vanilla extract for the vanilla paste if desired.
You can use whipping cream instead of heavy cream and light cream instead of half-and-half.
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