Preheat the oven to 300 degrees F.
Cut leeks in half lengthwise and slice into 1/2-inch pieces. Place leeks in a colander. Separate sliced leeks under cold running water and clean them. Drain well.
Heat a medium soup pot over medium high heat. Add 2 tablespoons extra-virgin olive oil, bay leaf and leeks. Season with salt and pepper. Peel potatoes and thinly slice them, adding them to soup pot as you go. Season potatoes with a little salt, add chicken stock to soup pot. Cover soup and bring to a boil. Uncover and simmer 12 to 15 minutes, until potatoes are very tender. Break up potatoes to slightly thicken soup. Remove bay leaf.
While soup simmers, rub the inside of a large mixing bowl with the cracked garlic. Slice the bread into small cubes. Add the bread to the bowl, drizzle generously with olive oil, and season well with salt and pepper. Toss the bread in the bowl to completely coat with oil, and pour onto a baking sheet in 1 layer. Place in the oven and toast until golden, about 6 or 8 minutes.
Fill a medium skillet with warm water and bring to a gentle simmer over medium low heat.
Meanwhile, cut slab bacon into 1/4-inch slices, making fat matchstick shapes. Cook bacon in small nonstick pan over medium high heat. Remove rendered bacon to paper towels. Return pan to heat and cook the shallots in the bacon dripping 3 to 5 minutes, until just tender, over medium heat. Whisk in 2 tablespoons of vinegar and Dijon until well combined. While whisking, slowly drizzle in the 1/4 cup olive oil until combined. Season with salt and pepper.
Pour the remaining tablespoon of vinegar into the simmering water. Crack an egg into a small bowl. Gently pour the egg into the simmering water. Repeat with the remaining 3 eggs. Cook eggs for about 2 minutes for runny yolks, or about 4 for solid yolks. Do not allow the water to boil! Carefully remove the eggs with a slotted spoon to a towel-lined plate to drain.
Tear the chicory into bite-sized pieces, and julienne the endive. Put the greens in the same bowl you tossed the bread in. Add the dressing, and tarragon and toss well with tongs. Divide between 4 plates, and top each with a poached egg. Season salad with more salt and cracked pepper.
To serve soup, ladle into bowls and top with bacon, croutons, chives and cracked pepper. Serve with crusty bread on the side.