Preheat the oven to 375°F (190°C) and place a rack in the middle position. Grease an 8x8-inch baking dish with butter or neutral oil. For cleaner removal, line with parchment paper and leave an overhang on two sides.
In a large bowl, whisk the milk and eggs until the eggs are fully broken up and the mixture looks uniform. Whisk in the maple syrup, melted butter, and vanilla until smooth. If the butter is very hot, cool it a few minutes first.
Sprinkle the baking powder, cinnamon, and salt over the wet mixture. Whisk for 15 to 20 seconds so the baking powder and spices disperse evenly without clumps.
Add the rolled oats and stir with a spatula until all oats are coated and no dry patches remain. Let the mixture rest for 5 minutes so the oats begin absorbing liquid, which helps the squares set after baking.
Add the fresh blueberries and fold them in gently with slow turns so most berries stay whole. This keeps visible bursts of blueberries throughout the pan and helps the mixture keep its color.
Pour the mixture into the prepared baking dish and spread into an even layer. Tap the dish on the counter once or twice to settle the batter. If blueberries are clustered, nudge a few toward sparse areas for even distribution.
Bake for 35 to 40 minutes, or until the edges look lightly golden and the center feels set when gently pressed. The surface should look less glossy and the middle should not jiggle when the pan is moved.
Cool in the pan on a rack for at least 20 minutes before slicing. Cooling allows steam to settle and helps the oatmeal firm up into sliceable squares. For very neat slices, chill for 30 minutes before cutting.
Slice into 9 squares for larger portions or 12 squares for smaller portions. Serve warm, room temperature, or chilled. Reheat individual squares with a tiny splash of milk to keep the texture soft.