Make the dilled crème fraîche:
In a small bowl, stir together the crème fraîche, dill, salt, and lemon juice, if using. Taste and adjust the seasonings; set aside.
Make the hash:
In a 10-inch cast-iron or other large heavy skillet, heat the butter and oil over medium-low heat. Add the potatoes, stirring to coat them with the fat. Cover and cook until the potatoes begin to turn golden and a bit tender with browned edges here and there, about 15 minutes. (you should hear only a faint sizzling). Uncover the potatoes and turn them with a spatula. Raise the heat to medium and cook until uniformly soft with some crisp brown spots, about 5 minutes. Stir in the onion and pat the mixture down with the spatula. Cook until the onion is soft and the mixture begins to stick in a few spots and browns on the bottom, about 5 minutes.
In a small bowl, whisk together the half-and-half, mustard, salt, and a generous grinding of pepper. Scrape up the hash and stir in the mustard mixture. Raise the heat to medium high. Continue cooking the hash, scraping it up and patting it back down another time or two until browned, another 5 to 8 minutes. Mix in the salmon, chives, and dill and cook until heated through, another 2 minutes.
Serve hot, topping each portion with a fried or poached egg (if using) and a sprinkling of capers. Garnish with dill sprigs or a bit of horseradish on the side, if you like. Top each portion with a dollop of the dilled crème fraîche, passing the rest at the table.