For the whole wheat bread: Place the whole wheat flour and the all purpose flour in a blender. In a bowl, add the yeast, the milk and the sugar. Incorporate the remaining ingredients to the flour. Knead until the dough becomes smooth and elastic.
Cover the dough and let it rest for 30 minutes. Shape the dough into balls of approximately 2½ ounces each.
Grease the pan with oil, place the balls on the pan and let them rise. Brush the tops with beaten egg. Bake at 400°F until slightly brown.
For the meat: Prepare the fire in the grill with charcoal. Thoroughly clean the grill. Wait until the fire is ready (no more black color on the briquets). Spread and distribute the charcoal to cook the skirt evenly.
Remove the meat from the refrigerator ahead of time, approximately 15 minutes before placing on the grill. Remove some of the external fat from the skirt. Add salt to taste.
Put the loin on the grill and cook for 12 minutes, 6 minutes each side. When the meat turns brownish, add a slice of cheese. Continue cooking until the cheese melts.
Concurrently, sweat the chopped onions with 1.5 tablespoons of olive oil until caramelized (approximately 10 minutes). Season with salt and fresh pepper.
Saute the mushroom slices with 1.5 tablespoons of olive oil and 1 crushed garlic clove (approximately 10 minutes). Season with salt and fresh pepper.
Spread mayonnaise or Dijon mustard over the bread. Add the cooked meat . Cover with sliced tomatoes, mushrooms and caramelized onions.