To make the dough, place the milk, 1/2 cup sugar, and salt in a saucepan over medium-high heat. Stir until the sugar melts and a few small bubbles appear at the edges of the pan. Remove from the heat source and let stand at room temperature.
Dissolve the yeast and remaining 1 teaspoon sugar in a bowl with the warm water. Stir with electric mixer or by hand with a wooden spoon until well blended. Let stand until bubbly, about 10 minutes.
Beat the eggs into the yeast mixture, using electric mixer on lowest speed. Stir in the cooled milk mixture (it should be lukewarm, about 98°F) and add the flour a little at a time, mixing until the dough begins to become elastic. Add the melted butter and stir again. Add more flour a little at a time until the dough is even more elastic and pulls away from the sides of the bowl.
Butter a large bowl with 1 tablespoon softened butter. Place the dough in the bowl and turn to coat. Cover with plastic wrap and let rise until doubled.
To make the cheese filling, combine all ingredients in a bowl. Beat with an electric mixer on high speed until smooth and thick. Set aside.
Preheat oven to 350°F. Line two large baking sheets with parchment paper.
To assemble the bread, punch down the dough and turn out onto a lightly floured surface. Pat the dough into a ball and cut 2 equal pieces. Roll out each piece to about 10x15 inches.
Spread half of the filling lengthwise down the center third of each rectangle. Cut 1-inch wide strips from each side of the filling out to the edges of the dough. Fold dough over the filling at each end, and then fold strips at an angle across the filling, alternating from side-to-side.
Use a large spatula or cake lifter to transfer the braids to the prepared pans. Brush the braids with the beaten egg yolk. Sprinkle each braid with Swedish pearl sugar.
Let the braids stand at room temperature until slightly puffy but not doubled, about 7-10 minutes.
Bake the loaves for 15-20 minutes, or until bread is golden brown and fragrant.
Serve warm or at room temperature.