Segment and juice 3 of the oranges in the following way: Cut both ends off 1 of the oranges and place it, cut side up, on a cutting board. Using a sharp, flexible knife, cut the skin and white membrane from the orange. Use a sawing motion and cut from top to bottom following the contours of the orange. Free the orange segments by cutting along the seams that separate them from each other. Do this over a bowl to collect any juices. Reserve the segments separately and squeeze what's left of the orange over the bowl to collect the juice. Do the same with the remaining 2 oranges. Strain the juice into a measuring cup: You should have about 3/4 cup; if not, juice the remaining orange and add as much as needed.
Combine the ginger, mayonnaise, mustard, and honey with 1/2 cup of the orange juice in a small bowl. Season with salt and pepper, to taste.
In a small bowl, combine half of the chopped shallots with 3 tablespoons of the remaining orange juice. Whisk in the olive oil and season with salt and pepper.
Cut the avocados lengthwise up to the large pit in the middle. Gently twist apart the 2 halves and remove the pit. Wedge a teaspoon between the skin and the flesh and remove the flesh in 1 piece. Cut the avocados into 1-inch pieces and put them into the bowl with the honey-and-ginger dressing. Add the remaining shallots, the chopped mint and the chopped cilantro, and mix together lightly so as not to break up the avocado.
When ready to serve, set aside 8 orange segments. Toss the rest with the whole lettuce leaves and just enough of the orange-and-olive oil vinaigrette to coat. Divide the lettuce and oranges among 4 plates. Spoon some avocado onto each plate (there will be some dressing left in the bowl). Top with the reserved orange segments and drizzle any remaining honey-and-ginger dressing over the salad. Garnish with whole mint and cilantro leaves, and, if you like, some pickled ginger.