Prepare the compote: Warm the butter in a small skillet over medium heat. Stir in the pine nuts and saute until lightly toasted, about 2 minutes. Watch the nuts carefully; they will continue cooking off of the heat and can burn easily. In a saucepan, combine the apricots, corn syrup, cinnamon, and almond extract with 1 cup of water. Bring to a simmer over medium heat, reduce the heat to low, and cook until the sauce is fairly thick and spoonable, about 10 minutes. Stir in the pine nuts. Keep the compote warm or let it cool to room temperature.
Start the pancake batter, placing 3/4 cup of the nuts in a food processor and pulsing briefly until ground. Avoid processing the nuts so long that they turn to butter. Add the flour, cornmeal, sugar, baking powder, and salt and pulse to combine just until a coarse meal forms. Spoon the mixture into a large bowl and stir in the butter until it disappears. Add the eggs, milk, almond extract, and remaining nuts. Warm a griddle or a large heavy skillet over medium heat. Pour a thin film of oil onto the griddle. Pour or spoon out the batter onto the hot griddle, where it should sizzle and hiss. A generous 3 tablespoons of batter will make a 4-inch pancake. Make as many cakes as you can fit without crowding. Cook the pancakes until their top surface is covered in tiny bubbles but before all the bubbles pop, 1 to 2 minutes. Flip the pancakes and cook until the second side is golden brown, 1 to 2 minutes longer. Repeat with the remaining batter, adding a bit more oil to the griddle as needed. Serve the pancakes immediately, accompanied with the warm compote.