Grill peppers on a grill, or on top of a gas range. Frequently turn peppers util evenly charred on all sides (about 5-7 minutes). Peel charred skin under cool running water. Cut peppers open; discard stems and seeds. Place peeled and seeded peppers in food processor. Cut out the stems of each tomato, then blanch and peel. Cut skinned tomatoes in quarters and squeeze out seeds from the tomato pieces. Place peeled and seeded tomatoes in the food processor.
Peel the cucumbers, and cut them in half the long way. Working with a teaspoon, scoop out the cucumber seeds. Cut the cucumbers into the coarse chunks, and add them to the food processors.
Place the bread in a bowl of cold water for 30 seconds, then squeeze the water out of the bread. Add the bread to the food processor, along with the garlic, onion, olive oil and balsamic vinegar. Process for at least 2 minutes, or until a smooth puree is achieved.
Place the gazpacho in a covered bowl and refrigerate for 2-4 hours (the longer it sits, the better the flavors will blend).
Working over a bowl, place the gazpacho in a strainer. Mash the gazpacho against the sides of the strainer with a wooden spoon, extracting every possible drop of liquid. This may take 15-20 minutes. Discard the content of the strainer. The gazpacho in the bowl will be medium-thin; if you 've used a very fine strainer, you'll have about 3 cups of gazpacho. Gently stir in the cucumber, eggs, and olives.
Peel the avocados, pit them, and cut into very coarse chunks. Drizzle chunks with lemon juice, to prevent discoloration, and season with salt. Gently stir into gazpacho.
Serve immediately in small soup bowls, garnished with parsley.