How to Make It
Remove and discard giblets from chicken. Place chicken, breast side up, on a clean work surface. Using a sharp knife, carefully cut on either side of the breastbone, cutting along the ribs to debone the breast halves without detaching them from the bird. Using kitchen shears, remove the backbone, separating the chicken into 2 halves. Discard ribs and breastbone; reserve backbone. Using a knife, remove the last 2 wing joints of each wing, leaving the first joint attached to the breast; reserve wing pieces. Sprinkle chicken halves all over with salt; set aside.
Heat 1 tablespoon oil in a medium saucepan over medium-high. Add chicken wing pieces and backbone; cook, stirring occasionally, until browned, 8 to 10 minutes. Add carrot, celery, and shallot; cook, stirring occasionally, until lightly browned, 3 to 4 minutes. Add wine; cook, scraping bottom of pan to loosen any browned bits, until liquid has almost completely reduced, 1 to 2 minutes. Add stock; bring to a boil over medium-high. Reduce heat to medium-low; cook, uncovered, until mixture reduces to about 3/4 cup, 25 to 30 minutes. Pour mixture through a fine wire-mesh strainer into a heatproof bowl; discard solids. Skim and discard fat from strained stock mixture.
While stock mixture reduces, heat remaining 2 tablespoons oil in a large (12-inch) cast-iron skillet over medium-high. Arrange chicken halves, skin sides down, in skillet. Sprinkle flesh sides of chicken evenly with pepper. Place a second heavy skillet on top of chicken; cook, undisturbed, until chicken is mostly opaque and skin edges are lightly browned, about 20 minutes. Remove top skillet from chicken; continue cooking, skin sides down, until skin is golden and releases from skillet, 5 to 8 minutes. Flip chicken halves; cook until a thermometer inserted in thickest portion of chicken just registers 155°F, 1 to 4 minutes. Let rest 10 minutes (internal temperature will rise to 165°F).
Return stock mixture to pan over medium-low. Add butter; cook, whisking constantly, until melted, about 2 minutes. Stir in parsley and lemon juice; season with salt to taste. Keep sauce warm over low until ready to serve. Serve chicken over cooked polenta with warm sauce.