In a large, heavy pot or an electric fryer, heat the vegetable oil (about 4 inches) to 360 degrees F. Fry the potatoes until golden brown, 3 to 4 minutes. Drain on paper towels. Season with salt and black pepper.
Heat the olive oil in a large saute pan over medium heat. Add the onions, season with salt and cayenne, and cook, stirring, for 2 minutes. Add the garlic and the salmon. Season with salt and cayenne and cook, stirring, for 1 minute. Add the potatoes, season with salt and cayenne, and cook for 2 to 3 minutes, stirring gently. Set aside and keep warm while you prepare the eggs.
Heat a very large nonstick skillet over medium-low heat. When hot, add the bacon drippings and heat until hot, 1 to 2 minutes. Add the eggs, 1 at a time, taking care not to break the yolks, and cook, undisturbed, until the white is set and the yolk is still runny, 2 to 3 minutes. Season the eggs with salt and black pepper, to taste. (If you wish the top of the eggs to be well-cooked, gently spoon some of the bacon drippings over the tops of the eggs as they cook.)
When ready to serve, spoon the salmon hash evenly among the centers of 4 plates. With a slotted spatula, lift the eggs out of the skillet, 1 at a time, and divide them among the plates, laying them gently over the top of the hash. Garnish with chives and serve immediately.