Make crêpe batter: Process eggs, flour, cornmeal, vanilla, 1 cup of the buttermilk, ⅓ cup of the sugar, and ¼ teaspoon of the salt in a blender until smooth, about 15 seconds. Cover; chill batter at least 2 hours or up to 8 hours (or overnight).

Cook blackberry syrup: Meanwhile, stir together blackberries, lemon juice, ⅓ cup of the sugar, and ¼ teaspoon of the salt in a small saucepan. Cook over medium, stirring occasionally, until berries begin to break down and mixture thickens slightly, 30 to 35 minutes. Remove from heat; cool at room temperature until ready to serve (at least 30 minutes or up to 4 hours).

Make blackberry cream: Beat heavy cream and remaining 1 ½ tablespoons sugar and ¼ teaspoon salt with a stand mixer fitted with a whisk attachment on medium speed until soft peaks form, about 2 minutes. Gradually add remaining ½ cup buttermilk, beating until medium peaks form, about 30 seconds. Cover and chill until ready to assemble crêpes (up to 4 hours).

Cook crêpes: Heat an 8-inch nonstick skillet over medium-low. Brush skillet lightly with ⅛ teaspoon of the oil; pour about 2 ½ tablespoons batter into skillet. Quickly tilt skillet in all directions so batter covers bottom of skillet. Cook until crêpe is almost set and can be shaken loose from skillet, about 1 ½ minutes. Flip crêpe, and cook 30 seconds. Remove from skillet. Repeat process with remaining oil and batter to yield 12 crêpes.

Assemble crêpes: Spread about 3 tablespoons buttermilk cream over 1 half of each crêpe. Fold crêpes into quarters, and arrange evenly on 6 plates. Top each serving with about ¼ cup blackberry compote. Garnish with powdered sugar; serve immediately.