Activate the Yeast: In a large bowl or stand mixer bowl, combine the warm milk and yeast. Let it sit for 5-10 minutes until foamy. Add the sugar, melted butter, egg, and salt. Mix briefly.

Make the Dough: Gradually add the flour, mixing on low speed until a shaggy dough forms. Knead by hand or with a dough hook for 6-8 minutes until the dough is smooth and elastic. Place in a lightly oiled bowl, cover, and let rise in a warm place for 1 hour, or until doubled in size.

Prepare the Filling: While the dough rises, combine the softened butter, brown sugar, and cinnamon in a small bowl. Mix until smooth. Set aside.

Shape the Rolls: Punch down the risen dough and turn it out onto a lightly floured surface. Roll the dough into a large rectangle, approximately 18x12 inches. Evenly spread the softened butter mixture over the dough, followed by a thin, even layer of apricot preserves.

Roll and Cut: Starting from the long edge, tightly roll the dough into a log. Slice the log into 12 equal pieces (about 1 inch thick). Arrange the rolls cut-side up in a greased 9x13 inch baking dish.

Second Proof and Bake: Cover the dish and let the rolls proof for another 30 minutes. Preheat your oven to 375°F (190°C) halfway through the proofing time. Bake for 20-25 minutes until golden brown. Let cool slightly.

Make the Glaze and Serve: Whisk together the powdered sugar, milk, and vanilla extract until smooth. Drizzle generously over the warm rolls before serving.