Prepare the leeks by placing the sliced pieces into a colander. Rinse them thoroughly, rubbing gently to remove any dirt. Repeat the rinsing process and allow the leeks to drain well. There is no need to dry them, as the residual water aids in cooking.
Heat a large Dutch oven or skillet over medium-high heat for 1 to 2 minutes. Add 2 tablespoons of olive oil to the heated pan, followed by the leeks. Season the leeks generously with salt and black pepper. Cook the leeks, while stirring occasionally, until the liquid has cooked off and the leeks start to stick to the pan, which should take around 5 to 6 minutes. Stir in the sliced garlic and fresh thyme. Decrease the heat to a low setting, cover, and cook for 20 minutes, stirring occasionally. Meanwhile, bring a medium pot of water to a boil over high heat. Carefully add the eggs and cook over medium-high heat for 7 minutes, ensuring to set a timer. Prepare an ice bath. Using a spider ladle or slotted spoon, transfer the eggs from the boiling water to the ice bath immediately. Allow them to cool for 3 to 4 minutes, then peel and set aside for later use.
Prepare the vinaigrette by whisking together miso paste, Dijon mustard, garlic, red wine vinegar, and olive oil in a large bowl. Uncover the leeks; they should have transformed into a silky, jam-like texture. Remove and discard the thyme sprigs. Transfer the leeks to the bowl, stirring to coat them evenly.
After adding the white beans, give it a few minutes to cool. Sprinkle in about ¾ of the parsley or tarragon leaves and toss to combine.
Divide the leeks and beans evenly among serving bowls. Drizzle each portion with a bit of olive oil. Place a halved soft-boiled egg on top of each serving and sprinkle with the remaining herbs. Optionally, serve with toasted bread on the side.