Rinse the livers under cold water, drain, and remove any white strands. Pat dry with paper towels. Combine the livers and port in a small salad bowl. Cover and refrigerate for 2 hours. Set a fine-mesh sieve over a second bowl and pour the livers and marinade into the sieve, reserving the marinade.
Wash the first bowl you used. Put the figs in the bowl, cover with hot water, and let stand as you go on with the recipe.
Melt 2 tablespoons of the butter in a large skillet over medium heat. Add the shallots and garlic and cook for 3 minutes, until soft and fragrant. Add the livers, thyme, bay leaves, salt, and pepper, and cook for 4 minutes, flipping the livers halfway through, until the livers are browned on the outside, but still pink inside. Add the reserved marinade, turn the heat up to medium-high, and cook for 6 to 8 minutes, until most of the liquids have evaporated and the livers are browned all over but still tender. Remove from heat, discard the bay leaves, and let cool for 5 minutes.
Transfer the liver mixture to a food processor and process until pureed. Add the remaining butter and pulse until smooth. Drain the figs, pat dry with paper towels, cut in 1/4 inch pieces, and fold into the liver mixture. Taste and adjust the seasoning.
Pack into two 1-cup glass jars, making sure there are no pockets of air. Place a small piece of plastic wrap directly on the surface of the terrine, close the jars tightly, and refrigerate for at least 6 hours, and preferably overnight.
Bring to just under room temperature and serve. The terrine will keep for up to 4 days, chilled, its surface covered with plastic wrap.