Prepare the vegetables by peeling and dicing the russet potatoes into 1/2-inch cubes, finely chopping the yellow onion, and dicing the red bell pepper. Mince the garlic and set everything aside in separate small bowls.
Chop the cooked beef brisket into small, bite-sized cubes so that it will crisp nicely in the skillet and distribute evenly throughout the hash.
Heat 1 tablespoon of the olive oil in a large heavy skillet over medium heat. Add the diced potatoes in an even layer and cook, stirring occasionally, for about 8 to 10 minutes until they begin to turn golden on the edges.
Add the remaining 1 tablespoon of olive oil and the unsalted butter to the skillet. Once the butter has melted, add the chopped yellow onion and diced red bell pepper. Cook for 4 to 5 minutes, stirring often, until the vegetables soften and the onion becomes translucent.
Stir in the minced garlic and cook for about 30 seconds, just until fragrant, making sure it does not brown.
Sprinkle the smoked paprika, ground black pepper, kosher salt, and dried thyme evenly over the potatoes and vegetables. Toss everything together so the seasonings coat the mixture thoroughly.
Add the chopped cooked beef brisket to the skillet, spreading it out over the potatoes and vegetables. Gently press the mixture down with a spatula to encourage browning and let it cook undisturbed for 3 to 4 minutes to develop a crispy crust on the bottom.
Stir the hash, flipping sections so that new areas come into contact with the hot skillet, and cook for another 4 to 5 minutes until the brisket edges are crisp and the potatoes are tender on the inside and well browned on the outside.
Use a spoon or spatula to make four shallow wells in the hash. Crack one egg into each well, keeping the yolks intact if you prefer runny centers.
Reduce the heat to medium-low, cover the skillet with a lid, and cook the eggs for 4 to 6 minutes, or until the whites are set and the yolks are cooked to your desired doneness.
Remove the skillet from the heat and sprinkle the chopped fresh parsley evenly over the brisket hash and eggs for a fresh, bright finish.
Serve the brisket hash and eggs straight from the skillet, scooping a portion of hash and one egg onto each plate while everything is hot and crisp.