Preheat the grill to medium-high for at least 15 minutes.
In a sauté pan over a stove, cook the bacon until crispy. Using a slotted spoon, remove the bacon from the pan, and set aside to drain. Beat together the eggs and season with salt and pepper. Scramble in the same pan as the bacon until just set. Remove from heat and set aside your grill.
In a large bowl, mix together the flour, baking powder, salt, and oregano. Add the shortening to the same bowl and cut the dry ingredients into the shortening using a pastry cutter or a fork. The idea is to work it until the dough is crumbly and in very small pieces. Add a handful or 2 of the bacon and toss gently to combine. Slowly add the milk, and stir the dough until it comes together, careful not to overstir it.
Using a rolling pin or your hands, spread the dough out into a large, round shape. Lightly grease 1 side of the dough with oil, and then grease the grill grates with a little oil as well. Place the dough on the grill, oiled side down, and cook until the dough is set, about 2 minutes. Flip the dough and top it with the scrambled eggs, remaining bacon, mushrooms, peppers, and cheese. Cook the pizza, covered, for about 10 minutes, or until the dough is cooked through and the cheese has melted.