In a small bowl, whisk together the flour, sugar, yeast, and salt.

Add the butter and, using your fingertips, rub it into the dry ingredients until the mixture resembles coarse bread crumbs.

In a small bowl, whisk together the milk and egg, then stir them into the flour mixture.

Press the dough until it comes together. (Add a little more liquid if the dough is not holding together.)

Place two large pieces of plastic wrap on a work surface. Turn the dough out and use the plastic wrap to to help press it into a square. Wrap well in the plastic and refrigerate for 1 hour.

On a floured surface, unwrap the dough and, working quickly so the butter doesn't melt, roll it out into a 12 x 18-inch rectangle. Fold like a letter again, then repeat the rolling-and-folding process at least four more times, one after the other.

Wrap the dough well in the plastic wrap and refrigerate for at least 4 hours or up to 4 days.

When the dough is chilled, generously butter a 12-cup muffin tin.

In a small saucepan, stir together the butter, brown sugar, maple syrup, cinnamon, and salt and heat over medium heat, stirring, until the butter and sugar have melted. Remove from the heat, stir in the walnuts, and let cool.

On a floured surface, roll out the Danish dough to a 12 x 18-inch rectangle. Spread the cooled filling evenly over the dough, leaving a 1-inch border all the way around.

Starting with the long edge, gently roll up the dough into a log. With a serrated knife, slice the roll into 2-inch-thick pieces and place one slice in each well of the prepared muffin tin, cut-side up. Cover the dough and let rise in a warm place until puffed up, about 45 to 60 minutes.

About 20 minutes before the end of the rising time, preheat the oven to 375°F (190°C).

Carefully uncover the dough.

Bake for 20-25 minutes, until golden brown.

Warm the maple syrup and butter together until the butter melts. Set aside.

Remove the buns from the oven, leaving them in the pan. Generously brush the warm buns with the maple-butter topping. Let cool for 15 minutes, then carefully remove the buns from the pan and place them on a baking sheet to cool completely. Spoon any walnut filling that has spilled out back into the buns.

These buns are best eaten the day they are made, but you can store any leftovers in an airtight container at room temperature overnight. Rewarm in the microwave, or in a 300°F (150°C) oven for 10 minutes.