How to Make It
Prepare a large bowl of ice water. In a large pot of boiling salted water, cook the leeks, covered, until tender, about 5 minutes. Using a slotted spoon, transfer the leeks to the ice water to cool. Drain well, then transfer the leeks to paper towels to dry, cut side down.
In a small bowl, whisk the vinegar with the shallot, mustard and tarragon. Slowly whisk in 1/4 cup plus 2 tablespoons of the olive oil. Season the vinaigrette with salt and pepper.
In a large skillet, heat the remaining 1/4 cup of olive oil. Working in batches, fry the speck over moderately high heat, turning once, until crisp around the edges, about 3 minutes. Drain on paper towels.
Add the butter to the skillet and reduce the heat to moderate. Crack 2 eggs into the skillet and fry, basting the eggs with fat, until the whites are set and the yolks are still slightly runny, about 3 minutes. Transfer the fried eggs to a plate. Repeat with the remaining eggs. Season the fried eggs with salt and pepper.
In a large bowl, coat the leeks with the vinaigrette, keeping them intact. Arrange the leeks on a platter or on individual plates and top with the speck and fried eggs. Drizzle any remaining vinaigrette on top, sprinkle with the chives and serve.
Speck is available at specialty food stores. If you can't find it, use regular prosciutto instead.