In a mortar or mini processor, work the basil and garlic together to form a paste. With the machine on, add 3 tablespoons of the oil in a steady stream until the mixture is smooth. Add the salt and Parmesan and process to blend. Scrape the pesto into a small bowl and pour the remaining 1 tablespoon oil on top; cover and refrigerate.
Roast the red bell peppers over a gas flame or under the broiler until charred all over. Transfer the peppers to a paper bag and let steam for 10 minutes. Discard the skin, cores, ribs and seeds and halve each pepper. Put the peppers in a shallow bowl and add the garlic and 2 tablespoons of the oil. Let marinate in the refrigerator for 2 hours.
Using a mandoline, cut the celery root into very thing strips, about 1/2 inch wide; alternatively, use a knife to cut it into fine julienne strips. Transfer to a large bowl and toss with the lemon juice. In a medium bowl, combine the pesto and mayonnaise. Add the celery root, season with salt and pepper and toss well.
Bring a large saucepan of water to a boil. Plunge the lobsters in head first, then cover and cook until they're red all over, about 10 minutes; transfer to a bowl and let cool.
Detach the lobster tails from the bodies. Using shears, cut through the underside of the tail shells and remove the meat. Halve the tails lengthwise and discard the intestinal veins. Twist off the claws, crack them and remove the meat. Transfer the lobster meat to a bowl.
Mound the celery root salad on 4 plates. Toss the lobster with the remaining 1 tablespoon oil, season with salt and pepper and arrange on top of the celery root. Drain the oil from the roasted peppers into a small bowl. Scrape of the garlic from the peppers and set a pepper half on each plate. Stir the vinegar into the pepper oil, season with salt and pepper and spoon around each salad. Garnish with basil and serve.
The pesto and peppers can be refrigerated separately overnight. Let the peppers return to room temperature before serving.
Most lobster dishes call for crisp whites, but the roasted peppers make a light red possible. Try an Italian or Californian Merlot.