Heat 4 tablespoons oil in a heavy large skillet over medium heat.
Add onion, anchovies, and garlic. Sauté until onion is tender but not brown, mashing anchovies with a fork, about 3 minutes.
Increase heat to medium-high. Add chicken livers and sauté until brown, about 4 minutes.
Add wine and cook until most of the liquid evaporates, about 3 minutes.
Add chicken broth and sage. Simmer until chicken livers are cooked through and liquid is reduced by half, breaking up livers with a fork, about 10 minutes. Cool slightly.
Transfer mixture to processor. Using short bursts, the process just until livers are coarsely pureed. Mix in lemon peel—season to taste with salt and pepper.