• In a large heavy-bottom skillet set over medium heat, add the bacon pieces and cook until crisp, stirring often. Transfer to a paper-towel lined plate to drain. 
• Using a small dish, pour off all but 2 tablespoons of the accumulated bacon dripping and reserve.
• Add the chopped onions and mushrooms to the skillet and continue cooking over medium heat until the become onions become translucent. Add the red bell pepper. Cook and stir until the bell pepper is heated through. Transfer the vegetables to a bowl and reserve. 
• Add 2 tablespoons of the reserved bacon drippings to the skillet, then add the sliced potatoes. Sprinkle the potatoes with salt and pepper and toss together. Add 2-3 tablespoons of water to the skillet, cover and cook for 15-20 minutes, stirring half-way through the process until tender and nicely brown.
• Add the reserved bacon and vegetables to the potatoes and stir to mix well. Toss with the chopped tarragon, reserving 1 tablespoon for final garnishing. 

Meanwhile, poach the eggs:

• Fill a saucepan with 4-inches of water and bring to a boil. Reduce the heat to low and keep the water at a low simmer for poaching the eggs.
• Crack the eggs, one at a time into a fine-mesh strainer. Swirl the egg around for a couple seconds to drain off the watery liquid. Transfer each egg into small dishes and tip into the simmering water. Cook for 3 to 3½ minutes or until desired doneness.  To serve:
• Divide the potato hash equally between four plates, place a poached egg on each serving and garnish with additional chopped tarragon.