Heat oven to 250 degrees. Spread bread cubes in an even layer on 2 rimmed baking sheets. Bake until edges have dried, but centers still slightly moist (cubes should yield to pressure)., 45-60 minutes, stirring several times during baking. Transfer dried bread to large bowl and increase oven to 375.
While bread is drying, use a paring knife to poke each chicken segment 4-5 times. Heat oil in a 12 inch skillet over med-high heat, until shimmering. Add wings to skillet and cook until golden brown on both sides, about 8 minutes. Transfer wings to separate bowl and set aside.
Melt the butter in the fat left in the skillet over med heat. Add onion, celery and 1/2 tsp salt and cook, stirring occasionally, until vegetables are softened (7-9 minutes). Stir in thyme, sage and pepper and coom until fragrant, about 30 seconds. Stir in one cup broth, scraping up any browned bits, and bring to a simmer. Remove from heat, add this vegetable mixture to the bowl with the dried bread, toss to combine.
Grease a 13x9 inch baking dish. Whisk the eggs, remaining 2 cups chicken broth, 1 1/2 tsp of salt and any accumulated juices from the wings in a bowl. Add this to the bread mixture withe the chopped parsley and toss to combine. Transfer to the prepared baking dish. Arrange wings on top of stuffing, cover tightly with foil, and place baking dish on a rimmed baking sheet.
Bake on lower rack until wings register 175 degrees, about 60 minutes.
Remove foil and transfer wings off stuffing to reserve for another use (soup or salad maybe). Gently fluff stuffing with fork, let rest 5 minutes before serving.