Combine whole wheat flour, undissolved yeast, sugar and salt in a large mixer bowl. Heat milk and butter until very warm (120° to 130°F). Add to bowl with egg. Beat for 3 minutes, scraping bowl occasionally. Gradually add all-purpose flour until dough forms a soft ball. Knead on a lightly floured surface until smooth and elastic, about 8 to 10 minutes. Cover; let dough rest for 10 minutes.
Combine mayonnaise and mustard until smooth. Roll dough into a 16 x 10-inch rectangle on a lightly floured surface. Spread mayonnaise mixture to within 1/2-inch of the edges. Sprinkle with ham and cheese. Roll up tightly from long side. Pinch seams to seal; shape into ring. Place on greased baking sheet. Cut 2/3 through ring at 1-inch intervals. Twist each section sideways. Cover; let rise in warm, draft-free place until doubled in size, about 1 hour.
Bake in a preheated 375°F oven for 25 to 30 minutes, or until lightly browned. Let cool 5 to 10 minutes before serving.