Adjust oven racks to upper-middle and lower-middle positions and heat oven to 250 degrees. Spread bread cubes in even layer on 2 rimmed baking sheets. Bake until edges have dried but centers are slightly moist (cubes should yield to pressure), 45 to 60 minutes, stirring several times during baking. (Bread can be toasted up to 1 day in advance.) Transfer to large bowl and increase oven temperature to 375 degrees.
Use tip of paring knife to poke 10 to 15 holes in each wing segment. Heat oil in 12-inch skillet over medium-high heat until it begins to shimmer. Add wings in single layer and cook until golden brown, 4 to 6 minutes. Flip wings and continue to cook until golden brown on second side, 4 to 6 minutes longer. Transfer wings to medium bowl and set aside.
Return skillet to medium-high heat and add sausage; cook, breaking sausage into ½-inch pieces with wooden spoon, until browned, 5 to 7 minutes. Transfer sausage to paper towel-lined plate, leaving rendered fat in skillet.
Heat butter with rendered fat in skillet over medium heat. When foaming subsides, add onion, celery, and ½ teaspoon salt. Cook, stirring occasionally, until vegetables are softened but not browned, 7 to 9 minutes. Add thyme, sage, and pepper; cook until fragrant, about 30 seconds. Add 1 cup broth and bring to simmer, using wooden spoon to scrape browned bits from bottom of pan. Add vegetable mixture to bowl with dried bread and toss to combine.
Grease 13 by 9-inch baking dish with butter. In medium bowl, whisk eggs, remaining 1 ½ cups broth, remaining 1 ½ teaspoons salt, and any accumulated juices from wings until combined. Add egg/broth mixture, dried figs, pecans, and sausage to bread mixture and gently toss to combine; transfer to greased baking dish. Arrange wings on top of stuffing, cover tightly with aluminum foil, and place baking dish on rimmed baking sheet.
Bake on lower-middle rack until thickest part of wings registers 175 degrees on instant-read thermometer, 60 to 75 minutes. Remove foil and transfer wings to dinner plate to reserve for another use. Using fork, gently fluff stuffing. Let rest 5 minutes before serving.