In a 2- to 3-quart pan over medium-high heat, melt butter. Stir often until butter is deep golden brown with darker brown flecks, 5 to 8 minutes (if butter foams up in pan, remove from heat briefly until foaming subsides). Pour butter into a 1-cup glass measure; chill until solid, about 1 1/2 hours.
Meanwhile, in a bowl, combine figs, orange juice, and 1 cup boiling water. Soak until figs have softened, about 30 minutes. Lift 1/2 cup figs from bowl and coarsely chop. Pour remainder, with soaking liquid, into a blender or food processor and whirl until smooth.
Scoop 1/2 cup solidified brown butter into a large bowl (bring to room temperature if necessary before using; reserve remainder for brown-butter sauce). With a mixer at medium speed, beat butter with granulated sugar until well combined. Add eggs one at a time, beating well after each addition. Add puréed figs and beat until smooth.
Add flour, baking powder, cinnamon, and salt and beat at low speed until smooth, scraping down sides of bowl as needed. Stir in chopped figs.
Scrape batter into a large buttered steamed pudding pan or an 11-quart bundt pan; Place lid on or cover tightly with foil and secure with a rubber band. Place pudding pan in a 12- by 17-inch baking pan and set on bottom rack of a 350° oven. Carefully pour 2 inches boiling water around the pan, then cover entire baking pan with foil.
Bake until pudding feels firm to touch (peel back foil to check) and a wooden skewer inserted in the center comes out clean, about 1 hour and 15 minutes. Let cool 10 minutes, then invert pan over a plate to unmold pudding. Let cool until barely warm. Cut into slices and serve with brown-butter hard sauce.
Brown Butter Sauce
Combine reserved brown butter, powdered sugar, orange emulsion & vanilla extract and beat until light and fluffy. You can serve at room temperature. Best if the Pudding is warm so the butter slightly melts.