A steamed pudding is crowned by a jammy topping of pureed dried apricots cooked in Armagnac.
Set a round wire rack in bottom of large stockpot. Set a 5-cup pudding basin or ovenproof bowl on rack. Fill pot with enough water to come about three-quarters of the way up the sides of the basin. Remove basin; dry, and butter inside. Set aside. Cover pot, and bring to a boil.
Put apricots, brandy, and 1/2 cup water into a small saucepan. Bring to a boil over medium-high heat. Reduce heat to medium-low, and simmer, partially covered, until apricots are very soft and most of the liquid has been absorbed, about 10 minutes. Reserve 1/2 cup apricots. Carefully puree remaining apricot mixture in a food processor (you should have about 1/2 cup); let cool 10 minutes.
Cut reserved apricots into 1/2-inch pieces; return to saucepan. Add 1/4 cup sugar and 1/3 cup water. Bring to a simmer; cook until liquid is nearly evaporated. Pour mixture into prepared basin, and set aside.
Butter a 10-inch round of parchment paper; set aside. Whisk flour, baking powder, and salt in a bowl; set aside.
Put remaining 3/4 cup sugar and the lime zest into the bowl of an electric mixer fitted with the paddle attachment. Mix on medium speed 30 seconds. Add butter; mix until pale and fluffy, about 4 minutes. Add eggs, one at a time, mixing after each addition. Mix in lime juice.
Add flour mixture in 3 additions, alternating with the apricot puree and milk. Transfer batter to prepared basin.
Place parchment round, buttered side down, over basin. Make a pleat in center of parchment. Cover with an 11-inch round of foil. Make a pleat in the center of the foil to allow room for pudding to expand. Cut a piece of twine about 7 feet long. Wrap twine twice around basin over foil, just below lip. Knot to secure. Tie loose ends to twine on other side of basin, creating a handle.
Lower pudding into boiling water; cover pot. Return to a boil; reduce to a simmer. Steam until an instant-read thermometer inserted into pudding registers 180 degrees, about 1 1/2 hour, adding boiling water occasionally to maintain level.
Transfer pudding to a wire rack. Remove foil and parchment. Let cool 10 minutes. Run a knife around edge of bowl to loosen, and invert pudding onto a serving plate. Serve warm, with ice cream or creme anglaise if desired.