Fill a small bowl with ice and water and set aside. Bring a small saucepan filled with water to a boil over high heat. Use a slotted spoon to slowly lower the eggs, one at a time, into the water. Cook for 7 minutes, turn off the heat, and use the slotted spoon to transfer the cooked eggs to the ice water. Once the eggs are cool enough to handle, gently roll them on a counter to crack the shell, then peel it away (be careful, as the egg yolks are still semisoft in the center).
To a small bowl, add the shallots, mustard, capers, cornichons, vinegar and olive oil and whisk to combine. Season with salt and pepper. Add the tarragon, chives, chervil and lemon juice and stir to combine. Quarter the eggs and gently add to the gribiche. Divide the eggs gribiche between two small bowls.
In a small bowl, mash the avocado with the lemon juice, Tabasco, salt and black pepper. Spread the avocado mixture over the toast and sprinkle with espelette pepper. Divide four pieces of avocado toast between four plates and serve with the bowls of eggs gribiche on the side.