Place warm water in large warm bowl. Sprinkle in yeast; stir until dissolved. Add warm milk, butter, sugar, salt, and 2 cups all-purpose flour; blend well. Stir in 2 eggs, whole wheat flour, and enough remaining all-purpose flour to make soft dough. Grease top; cover tightly with plastic wrap. Refrigerate 2 to 24 hours.
Remove from refrigerator. Punch dough down. Remove dough to lightly floured surface. Divide dough half; set aside half. Divide remaining dough in half. Shape 1 portion into ball for body. Place on large greased baking sheet; flatten slightly. Divide portion in half; break off 3/4-inch piece, and shape into ball for muzzle. Shape remaining piece into ball for head. Attach head to body; pinch to seal. Attach nose to head; pinch to seal. Divide remaining half of portion into 6 equal pieces. Shape into balls, and attach to body and head to form paws and ears; pinch to seal. Press date pieces or raisins into dough for eyes, nose and buttons. Repeat with remaining half of dough to make second bear. Cover; let rise in warm, draft-free place until doubled in size, about 40 to 60 minutes.
Beat remaining egg with 1 tablespoon water; brush over dough. Redefine shapes, if necessary. Bake at 350oF for 25 minutes or until golden. Remove from sheets; let cool on wire racks.