In a mixing bowl, combine all-purpose flour, granulated sugar, instant yeast, the room-temperature egg, warm milk, and egg yolk. Mix until just combined, then knead in softened unsalted butter until the dough becomes smooth and elastic, about 5-7 minutes. Cover the bowl with a kitchen towel and place it in a warm area for 1-2 hours, or until the dough has doubled in size.

While the brioche dough is rising, make the cream cheese filling. In a mixing bowl, beat cream cheese until creamy and smooth, approximately 2-3 minutes. Gradually add powdered sugar, milk, and vanilla bean paste, continuing to mix until the mixture is light and fluffy. Set this filling aside while the dough finishes rising.

In another bowl, combine brown sugar, flour, ground cinnamon, maple syrup, and melted butter. Stir until the mixture resembles coarse crumbs, about 1-2 minutes. Set it aside until you are ready to assemble the danishes.

Once the brioche dough has risen, punch it down gently to release air and divide it into 8-12 pieces. Roll each piece into a ball, flatten them into circles with raised edges. Place a generous dollop of the cream cheese filling and a spoonful of pumpkin butter in the center of each circle. Sprinkle the coffee cake crumb generously over the top.

Preheat your oven to 375°F (190°C). Transfer the shaped danishes to a parchment-lined baking sheet, ensuring space for expansion. Brush the edges with a beaten egg for a golden finish. Bake for 20-30 minutes until golden brown and puffed.

As the danishes cool for about 10 minutes, prepare the glaze. Whisk together powdered sugar, vanilla extract, milk, and a pinch of salt until smooth and no lumps remain. Adjust the consistency by adding more milk or powdered sugar, then drizzle over the slightly warm danishes.