Butter an 8-inch square baking dish; set aside. Put 3 tablespoons of the butter in a microwave-safe small bowl or liquid measure; microwave for 30 seconds to 1 minute to melt the butter.
Whisk together the flour, baking powder, baking soda, 3 tablespoons of the sugar and 1/2 teaspoon salt in a large bowl. In a separate small bowl, whisk together the buttermilk, 2 of the eggs and the melted butter. Add the wet ingredients to the dry ingredients, and stir until just combined.
Heat a large nonstick skillet or griddle over medium heat. Melt a little of the remaining butter in the pan. For each pancake, spoon 1/4 cup of batter into the skillet and cook until the bottom is golden brown and small bubbles appear on the top, 2 to 3 minutes. Flip and cook until golden brown on the other side, about 2 minutes. If the pancakes brown too quickly, reduce the heat to medium-low. Transfer the pancakes to a plate as finished. Repeat until all the batter is used, adding more butter to the pan as necessary. (You should have at least 12 pancakes.)
Whisk together the half-and-half, vanilla, the remaining 5 eggs and 3 tablespoons sugar, and 1/4 teaspoon salt in a large bowl.
Shingle the pancakes in the prepared baking dish. Pour the egg mixture evenly over the pancakes. Cover the dish with plastic wrap and refrigerate for at least 8 hours up to overnight to allow the pancakes to soak in the custard.
Preheat the oven to 350 degrees F. Remove the plastic wrap, and bake the casserole until it feels firm yet springy to the touch in the center and the custard is set, 50 to 60 minutes. Let cool 15 minutes before serving. Serve warm with maple syrup, blueberries and pats of butter.
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