Prepare the vegetables by finely chopping the yellow onion and dicing the red bell pepper into small, even pieces so they cook quickly and evenly.
Heat a large skillet over medium heat and add the olive oil, allowing it to warm until it shimmers but does not smoke.
Add the chopped onion to the skillet and sauté for 3 to 4 minutes, stirring often, until the onion becomes soft and translucent.
Stir in the diced red bell pepper and continue to cook for another 3 to 4 minutes, until the pepper begins to soften and the edges take on a light golden color.
Add the minced garlic to the skillet and cook for about 30 seconds, just until fragrant, being careful not to let it brown.
Add the shredded or chopped cooked beef brisket to the skillet, breaking up any large pieces with a spatula so it spreads evenly among the vegetables.
Season the brisket mixture with smoked paprika, ground black pepper, salt, dried thyme, and red pepper flakes if using, then stir well to coat everything with the spices.
Cook the brisket and vegetable mixture for 5 to 6 minutes, stirring occasionally, until the brisket is heated through and lightly crisped in a few spots.
Spread the brisket mixture into an even layer in the skillet and reduce the heat to medium-low to keep it warm while you cook the eggs.
In a separate nonstick skillet, melt the unsalted butter over medium heat until it foams gently but does not brown.
Crack the eggs into a small bowl one at a time to check for shells, then gently slide them into the buttered skillet, spacing them apart.
Cook the eggs to your preferred doneness, allowing the whites to set while the yolks remain runny for sunny-side up, or flipping gently for over-easy or over-medium if desired.
While the eggs cook, taste the brisket mixture and adjust the seasoning with a little more salt or black pepper if needed, then keep it warm over low heat.
Divide the hot brisket and vegetable mixture evenly among four plates, forming a generous bed for each serving.
Top each portion of the brisket mixture with one cooked egg, placing it in the center so the yolk sits proudly on top.
Sprinkle the chopped fresh parsley and sliced chives evenly over the plated brisket and eggs to add color and a fresh, herbal finish.
Serve the brisket and eggs breakfast immediately while hot, encouraging everyone to break into the yolk so it mingles with the savory brisket and vegetables.