Preheat the oven to 375 degrees F. Whisk the eggs, heavy cream, chives, 3/4 teaspoon salt and a few grinds of pepper in a bowl until fluffy. Generously coat the bottom and sides of an 8-inch-square or round cake pan with olive oil; add the egg mixture. Set in a roasting pan; add enough water to come 1/4 inch up the sides of the cake pan. Carefully transfer to the oven; bake until the eggs are set, 20 minutes.
Meanwhile, heat 2 tablespoons olive oil in a large skillet over medium-high heat. Working in batches if needed, add the Canadian bacon and cook until crisp, 2 to 3 minutes per side; set aside.
Cut the baked eggs into 4 equal pieces. Spread the rolls with the dijonnaise, then sandwich with the Peppadews, bacon, cheese and eggs. Place the sandwiches on a baking sheet and bake until the cheese melts, about 3 minutes.
Meanwhile, toss the greens with the remaining 1 tablespoon olive oil, the Peppadew brine and fried shallots; season with salt and pepper. Serve with the sandwiches.