Trim chicken of excess fat, but leave skin intact. Cut chicken into 8 pieces. Pat the chicken dry and sprinkle with salt and pepper.
Heat 2 tsp oil in a large skillet over medium-high heat. When shimmering and almost smoking, add chicken pieces, skin side down. Cook, undisturbed, 5-8 min or until golden brown.
Flip chicken over. Reduce heat to medium-low. Add 3/4 c broth, cover tightly and simmer 10-16 min, until breasts register 155°F and thighs register 170°F on instant read thermometer inserted into thickest part. Remove chicken to a plate, skin side up.
Drain liquid from skillet into Pyrex measuring cup. Wipe skillet clean with paper towel. Add 1 tsp oil to skillet and heat until oil is again shimmering. Return chicken to pan, skin-side down, and cook 4-7 min, until skin is deep golden brown, and breasts read 160°F and thighs 175°F Remove chicken to serving platter, skin-side up, and cover loosely with foil.
Skim fat from reserved liquids and add enough broth to make 3/4 c total.
Add 1 tsp oil to skillet (do not wipe clean this time) over low heat. Add shallot and cook, stirring often, until soft, 1 minute. Add flour and cook 30 seconds, stirring constantly. Increase heat to medium-high, add liquid, and bring up to a simmer, scraping the fond off the bottom of the pan with a wooden spatula or spoon. Keep at a rapid simmer 2-3 min, or until reduced to 1/2 cup. Add any juices accumulated in the serving platter.
Remove from heat and whisk in lemon juice, parsley, chives and butter. Adjust seasoning to taste, pour sauce around chicken, and serve.