Bring 4 cups water to a boil in a medium saucepan, add the squares of meat and skim off any grayish foam that rises to the top during the first few minutes of boiling. Add the garlic, bay leaves, herbs, a generous 1 teaspoon salt and half of the onion. Simmer over medium to medium-low heat for an hour or so, until the meat is tender. If there is time, let cool in the broth. Drain and discard all but the meat; then shred the meat into thin strands.
Combine the cooked meat, the remaining half of the onion, the radishes, chile, tomato and cilantro, and toss with the sour orange or lime juice and generous 1 teaspoon salt. Line a platter with 4 of the romaine leaves; slice the remaining 2 to make a bed in the center. Scoop the dzik into a mound over the sliced lettuce. Decorate with slices of avocado.