Place the cheese in some cheesecloth or several layers of paper towels and let it drain for 5 to 10 minutes, pressing gently, to remove excess moisture.
With a mixer, beat the cheese until smooth. Add the egg and continue to beat until incorporated. Sprinkle in the sugar and salt, and then gradually add the flour; continue to mix until a dough begins to form. Add the melted butter and mix just until the dough holds together when patted with your hands, it will look pretty wet. Wrap the dough in plastic and refrigerate for several hours or up to overnight, the dough will stiffen as it chills.
Using a knife, make a slit down the seam of the apricot and carefully pull out the pit, keeping the fruit in tact. Stick a sugar cube inside the apricot where the pit was and squeeze the apricot back together to close it up tightly. The sugar cube melts inside the apricot when you cook it.
Bring a large pot of water to a boil. Lightly flour a rolling pin and work surface. Roll the dough out into a thin circle and cut out 12 circles with a 4-inch ring cutter. Place an apricot in the center of the dough and carefully bring the edges together to completely enclose the fruit and form a dumpling; make sure there are not any tears or holes in the dough. Working in batches, gently lower the dumplings into the simmering water; they should not be touching or crowded. Cook for about 10 to 15 minutes; the dumplings will sink to the bottom of the pot and then rise to the surface. Carefully remove the dumplings from the water with a strainer.
In a large nonstick skillet, melt the butter over medium heat. Add the bread crumbs, sugar, and cinnamon, cook and stir for a couple of minutes until the bread crumbs are toasty and golden. Lay the bread crumbs out on a sheet pan or plate and roll the apricot dumplings in the browned crumbs to coat evenly.
In a small bowl, mix the sour cream, brown sugar, and lemon juice together. Serve the apricot dumplings with the sweet-sour cream sauce on the side and a dusting of confectioners' sugar if desired.