Preheat oven to 350°F. Position a rack in the center and place a baking sheet on the lower rack to catch drips.
Make the caramel topping: Place 1/4 cup butter in your cake pan and put it in the oven for 2-3 minutes until melted. Remove carefully, then sprinkle 1/2 cup brown sugar evenly over the melted butter. Stir gently with a fork until smooth and combined. Set aside to cool for 3 minutes.
Prep the berries: In a small bowl, toss the blueberries with 1 tablespoon of flour until lightly coated. Layer them evenly over the caramel mixture in the pan.
Whisk dry ingredients: In a medium bowl, whisk together 1 1/2 cups flour, baking powder, baking soda, and salt. Set aside.
Cream butter and sugar: In a large bowl, beat 1/2 cup softened butter and 3/4 cup granulated sugar together until light and fluffy, about 2-3 minutes. Add egg and egg yolk one at a time, beating well after each. Beat in vanilla.
Add dry and wet alternately: With mixer on low, add half the dry ingredients, then half the buttermilk, then the rest of the dry, then the rest of the buttermilk. Mix just until combined. Batter will be thick and scoopable.
Assemble and bake: Spoon batter gently over the blueberries in the pan. Spread evenly. Bake 40-45 minutes until deep golden brown and edges pull away from the pan.
Rest before flipping: Place pan on a cooling rack and let rest exactly 5 minutes. Set a timer. Do not skip.
Flip the cake: Place serving plate upside down over the pan. Holding together firmly with oven mitts, flip over. Let pan sit on top for 30 seconds without lifting. Then lift pan slowly. If a few berries stick, press them back into place with a finger.