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, brandy, 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 hard 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 buttered 11-cup bundt pan; cover tightly with foil and secure with a rubber band. Place pudding in a 12- by 17-inch baking pan and set on bottom rack of a 350° oven. Carefully pour 2 inches boiling water around bundt 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 bundt pan over a plate to unmold pudding. Let cool until barely warm, then sprinkle with powdered sugar. Cut into slices and serve with brown-butter hard sauce.
Per serving with about 2 tablespoons of brown-butter hard sauce: 445 cal., 30% from fat; 6 g protein; 15 g fat (5 g sat.); 75 g carbo (9 g fiber); 363 mg sodium; 113 mg chol.
Brown-butter hard sauce: In a bowl, with an electric mixer on medium speed, beat reserved solid brown butter from brown-butter figgy pudding (recipe precedes) until smooth. Add 1 cup powdered sugar and beat on low speed until well combined. Add 1 tablespoon brandy and 1 teaspoon vanilla; beat on high speed until light and fluffy, 3 to 5 minutes. Spoon into a bowl; serve at room temperature, or cover and chill up to 3 days. Bring to room temperature and beat again to restore texture before serving. Makes about 1 1/4 cups.
Per tablespoon: 66 cal., 62% from fat; 0 g protein; 6 g fat (9 g sat.); 6 g carbo (0 g fiber); 47 mg sodium; 12 mg chol.
You can brown the butter and make the hard sauce up to 3 days ahead; cover separately and chill, then bring hard sauce to room temperature before serving. You can make the pudding up to 1 day ahead; cool, cover, and store at room temperature.