Stir warm milk, 1 tablespoon sugar and yeast in the bowl of a stand mixer. Let stand until foamy, about 10 minutes. Add egg, remaining sugar, flour and salt; mix with the dough hook on low until a shaggy dough forms. Add butter a few cubes at a time, mixing until incorporated. Increase to medium and knead until smooth and elastic, 12 to 15 minutes. Transfer to a lightly buttered bowl, cover and let rise until slightly puffed, about 40 minutes.
Gently deflate the dough, roll into a rectangle and shape into a tight log to fit a 9-by-5-inch loaf pan. Place seam-side down in a well-buttered pan, cover and let rise until nearly doubled, about 30 minutes.
Heat oven to 340°F/170°C. Bake until deeply golden and set, about 30 minutes. Turn out and cool completely on a rack before slicing thickly.
Whisk eggs, milk, cinnamon and brown sugar in a shallow bowl. Melt a cube of butter in a nonstick skillet over medium heat. Dip each slice in custard, about 15 seconds per side, then cook until golden and crisp at the edges, 2 to 3 minutes per side, adding butter as needed.
Stack slices with small cubes of butter between them. Finish with banana, blueberries, confectioners’ sugar and maple syrup. Serve at once.