Lightly spoon flour into dry measuring cups; level with a knife. Combine 2 cups flour, beer, and yeast in a large bowl, stirring with a whisk. Cover and refrigerate 8 hours or overnight.
Remove mixture from refrigerator; let stand 1 hour.
Add 2 cups flour, 1 tablespoon granulated sugar, and salt; stir until a soft dough forms. Turn dough out onto a floured surface. Knead until smooth and elastic (about 8 minutes); add enough of remaining flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel tacky). Knead in cherries and chocolate.
Place dough in a large bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in size. (Gently press 2 fingers into dough. If indentation remains, dough has risen enough.).
Punch dough down; cover and let rest 5 minutes. Shape dough into a 9-inch round; place on a baking sheet lined with parchment paper. Lightly coat dough with cooking spray. Cover and let rise 1 hour or until doubled in size.
Preheat oven to 350°.
Uncover the dough. Combine water and egg white, stirring with a whisk, and brush over dough.
Sprinkle the dough with pearl sugar, if desired. Make a 1/4-inch-deep cut down the center of dough using a sharp knife.
Bake at 350° for 30 minutes or until bread is browned on bottom and sounds hollow when tapped. Remove from pan; cool on a wire rack.