Make the ice cream: Carefully cut kernels from cobs; transfer to a saucepan. Break cobs in half; add to saucepan. Stir in milk, cream, 1/2 cup granulated sugar, and 1 teaspoon salt. Bring to a boil. Let cool; discard cobs.
Working in batches, puree corn mixture in a blender until smooth. (For safety, remove cap from hole in lid, and cover with a dish towel to prevent spattering.) Return mixture to saucepan, and bring to a simmer over medium heat. Remove from heat.
Meanwhile, prepare an ice-water bath. Whisk together
egg yolks and remaining 1/4 cup granulated sugar in a small bowl. Whisk 1 cup corn mixture into yolks, then return entire mixture to saucepan, whisking constantly. Cook over medium-low heat, whisking constantly, until custard thickens and can easily coat the back of a spoon, about 10 minutes.
Strain custard through a fine sieve into a bowl, pressing down solids; discard solids. Transfer bowl to ice-water bath, and refrigerate for 1 hour.
Freeze in an ice cream maker according to manufacturer's instructions. Transfer to an airtight container, and freeze until firm, at least 3 hours.
Meanwhile, make the shortcakes: Preheat oven to 400 degrees. Sift flours, baking powder, baking soda, granulated sugar, and
1 1/4 teaspoons salt twice into a bowl. Cut in butter using a pastry cutter or rub in with your fingers until small clumps form. Make
a well in the center, and pour in buttermilk. Stir until combined.
Turn out dough onto a generously floured surface, and fold over onto itself 3 or 4 times. Roll out dough to a 1-inch thickness. Cut out rounds of dough using a 3-inch biscuit cutter, and transfer to a parchment-lined baking sheet. Gather scraps, and repeat once. Brush shortcakes with heavy cream, and sprinkle with sanding sugar. Bake until golden brown, 18 to 20 minutes.
Make the blueberry compote: Bring 1 cup blueberries, the granulated sugar, water, and lemon juice to a simmer in a small saucepan over medium heat. Cook until berries burst and liquid thickens slightly, about 5 minutes. Stir in remaining 1/2 cup blueberries. Let cool.
Split shortcakes in half, and top each shortcake bottom with a spoonful of blueberry compote, top with a scoop of ice cream, then sandwich with shortcake tops.
Ice cream can be frozen for up to 3 months. Shortcakes can be stored at room temperature for up to 3 days. Compote can be refrigerated for up to 3 days.