FOR THE DOUGH: Lightly grease a large bowl with cooking spray or neutral oil. Mix the butter, milk, sugar, and egg yolks together in a large measuring cup. Sift 2½ cups flour, the yeast, and salt together in the bowl of a standing mixer fitted with the dough hook. With the mixer on low, add the milk mixture. After the dough comes together, increase the speed to medium and mix until shiny and smooth, 4 to 6 minutes. (If the dough is sticky after 3 minutes, add the remaining ½ cup flour, 2 tablespoons at a time.) Turn the dough out onto a heavily floured surface and shape into a ball. Knead just until the dough becomes smooth, about 20 seconds. Place the dough in a greased bowl, cover with plastic wrap, and let rest in a warm place until doubled in size, 1½ to 2 hours.
FOR THE FILLING: While the dough is rising, bring the dates, milk, 3 tablespoons sugar, and cinnamon to a simmer in a small nonstick skillet medium-high heat (I actually used a small saucepan). Lower the heat to medium-low and cook until the milk is evaporated and the mixture becomes a thick paste, about 5 minutes. Cool to room temperature, at least 30 minutes.
With an electric mixer, beat the egg whites at medium-low speed until frothy, about 1 minute. Increase the speed to medium and gradually add the remaining ¾ cup sugar until incorporated, about 1 minute. Increase the speed to high and beat until the whites hold soft peaks. Finally, add cooled date mixture and beat on low speed until just incorporated, about 10 seconds.
Coat two 8-inch cake pans with cooking spray. Deflate dough, remove it from the bowl, and divide into two balls. Rolls each pieces of dough into a 20-inch square on a lightly floured surface.*** Spread each square with half the filling and top with half the walnuts. Roll each into a cylinder and arrange in the prepared pans in a spiral. Cover with plastic wrap coated with cooking spray and let rise 1 hour.
Adjust an oven rack to the middle position and heat oven to 350 degrees F. Brush the dough spirals with the beaten egg and sprinkle with the remaining 1 tablespoon of sugar. Bake until golden brown, 25 to 30 minutes. Serve warm (but cold is still awesome).
Notes
* Make sure you whip those egg whites properly. If not, the filling isn't quite as light and spongy. Adding the sugar slowly, very slowly, as the egg whites are being whipped is the key. And don't overbeat them.
** If the dates are not finely chopped, your date "paste" will be lumpy and harder to incorporate into the egg whites. I roughly chopped my dates on one trial and found the lumpiness not prohibitive to taste, but not so pretty for presentation. Learn from me: finely chop dates = pretty pretty.
*** My mom and I made 16 x 20-inch and a 16 x 17-inch rectangles, rolled very thinly, almost paper thin. The difference between the two dough sizes was that I eyeballed the dough balls when I halved the dough -- and obviously did not do so evenly.
Recipe by story of a kitchen at https://www.storyofakitchen.com/bread-recipes/potica/