Heat the olive oil in a frying pan and fry the bacon for 2 to 3 minutes over a medium–high heat until crisp. Remove with a slotted spoon and set aside. Fry the bread in the bacon fat until crisp, then drain on paper towels.
Put the lettuce in a salad bowl and add the bacon and crispy bread.
Mix the vinegar with the mustard and tarragon and plenty of seasoning, then whisk in the extra-virgin olive oil to form a vinaigrette.
Bring a small saucepan of water to the boil, then reduce to a gentle simmer. One at a time, crack the eggs directly into the simmering water; poach them for 2 to 3 minutes so the whites are cooked but the yolks still runny. Drain with a slotted spoon on paper towels.
Pour the vinaigrette over the salad and toss to combine. Divide between four plates and top each with a poached egg. Scatter with the chives and serve.