Heat the oil in a large skillet over low heat until hot. Add onion and cook, stirring occasionally, until just beginning to turn brown.
Remove onion to the bowl of a food processor. Add chicken livers and port to the skillet and cook 5 minutes turning on each side until lightly browned and liquid has reduced. Add cooked livers and their liquid, eggs, chicken fat, salt, and pepper to the onions and pulse several times until smooth, scraping down the sides if necessary.
Scoop liver mixture into serving bowl or mold, and refrigerate at least 2 hours. When ready to serve, scoop or unmold the chopped liver over lettuce leaves on a platter. Serve with crackers.