In a big stockpot, bring salted water and the vinegar to a boil over high heat.
Add in the lobsters and cover the pot.
Return to a boil and cook for 8-10 minutes, or until the shells are bright red.
Lift the lobsters from the pot with tongs and set them aside to drain and cool.
When cool enough to handle, crack the shells and remove the meat; discard the shells.
Transfer meat to a bowl, cover, and refrigerate.
In a small saucepan, cook the potatoes in boiling salted water over med-high heat for about 15 minutes, or until tender when pierced; drain and, when cool enough to handle, cut the potatoes into ½ inch dice; put them in a bowl and set aside.
In a small saucepan, cook the celery in boiling salted water for about 2 minutes, until bright green.
Using a slotted spoon, remove and rinse them under cold water to set the color; set aside.
Let the water return to a boil; add in the beet, decrease the heat, and simmer about 35 minutes, or until tender when pierced; drain and when cool enough to handle, peel and cut into ½ inch dice; set it aside.
Cut the avocados in half lengthwise and remove the pits; peel and cut avocados into quarters.
Using a sharp, thin knife, cut each quarter lengthwise into 1/8 inch slices, keeping the slices attached at one end.
Fan the slices gently and set them aside.
In a medium bowl, combine the lobster meat, potatoes, celery, and romaine.
Add the mayonnaise, 1 tablespoon at a time, until the salad is nicely dressed.
Season to taste with salt, white pepper, cayenne, and lemon juice.
Fold in tarragon and chives.
Divide the salad among 4 martini glasses or salad plates and garnish each with 2 avocado fans and a portion of the diced beets.