For the dressing: Mix the olive oil, sugar, parsley, basil, vinegar, salt, pepper, thyme, green onions and garlic in a large bowl.
For the tomatoes: Cut the tomatoes into slices, chunks, quarters or halves, depending on the size. Add the tomatoes to the dressing and marinate for 3 to 4 hours.
To serve: Lay out the butter lettuce leaves on a platter. Remove the tomatoes with a slotted spoon and spoon generously onto the lettuce. Drizzle over some of the dressing. Sprinkle over lots of whole basil leaves and basil stems with a few leaves on then.