Plunge lobsters headfirst into a 12-quart pot of boiling salted water. Cook, covered, over high heat 4 minutes (for 1 1/4-lb lobster) to 5 minutes (for 1 1/2-lb lobster) from time they enter water. Transfer with tongs to a large colander to drain and cool. (Lobsters will not be fully cooked.) Return cooking water to a boil.
Add potatoes to boiling water and simmer, uncovered, until just tender, 15 to 20 minutes. Drain in a colander.
When lobsters are cool enough to handle, twist off claws (including knuckles) and crack with a meat mallet or rolling pin. Halve lobsters lengthwise using a large heavy knife. Reserve green tomalley and any orange roe from bodies for another use if desired.
Prepare grill for direct-heat cooking over medium-hot charcoal (moderately high heat for gas).
While grill heats, whisk together vinegar, lemon juice, garlic, red-pepper flakes, sugar, and salt in a bowl. Add oil in a slow stream, whisking until well blended, then stir in basil.
Halve potatoes. Oil grill rack, then grill potatoes, cut sides down, covered only if using a gas grill, turning over once, until grill marks appear, about 4 minutes total. Transfer to a bowl and toss with 2 tablespoons vinaigrette. Keep potatoes warm, covered.
Oil grill rack, then grill 4 lobster halves, cut sides down, and all of claws, covered only if using a gas grill, until grill marks appear on meat, about 4 minutes. Turn over bodies and claws and grill in same manner 4 minutes more. Transfer to a large bowl and toss with 1/4 cup vinaigrette, then keep warm, covered. Grill remaining lobster halves in same manner and add to same bowl, tossing to coat.
Arrange arugula on a large platter and top with grilled lobster and potatoes. Drizzle with 1/2 cup vinaigrette and serve remaining vinaigrette on the side.
• Lobsters can be cooked in 2 batches in an 8- to 10-quart pot.
• If you aren't able to grill outdoors, lobsters and potatoes can be cooked in a hot oiled well-seasoned large (2-burner) ridged grill pan over moderately high heat.
Makes 4 servings.