Step 1

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.

Step 2

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.

Step 3

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.

Step 4

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.

Step 5

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.