Season chicken with salt and pepper; toss with 2 tablespoons pesto. Heat a large cast-iron skillet over medium-high. Swirl in 1 tablespoon oil. Add chicken in a single layer, skin-sides down; cook 2 minutes. Reduce heat to medium. Top with a parchment round, then a skillet weighted with canned goods; press down firmly. Cook 5 minutes. Flip chicken; return parchment and weighted skillet to top. Cook until a thermometer inserted in thickest parts of chicken reads 165 degrees, 6 to 7 minutes. Transfer to a plate; cover.

Transfer all but 2 tablespoons pan juices to a bowl. Add leeks to skillet over medium heat, season with salt, and cook until tender, 6 to 7 minutes. Stir in remaining 2 tablespoons pesto; cook 1 minute. Add chickpeas, 1 cup water, and reserved pan juices. Cook until slightly reduced, 6 to 8 minutes. Toss fennel with remaining 1 tablespoon oil; season with salt and pepper. Serve with chicken and chickpeas, garnished with fennel fronds.

Season chicken with salt and pepper; toss with 2 tablespoons pesto. Heat a large cast-iron skillet over medium-high. Swirl in 1 tablespoon oil. Add chicken in a single layer, skin-sides down; cook 2 minutes. Reduce heat to medium. Top with a parchment round, then a skillet weighted with canned goods; press down firmly. Cook 5 minutes. Flip chicken; return parchment and weighted skillet to top. Cook until a thermometer inserted in thickest parts of chicken reads 165 degrees, 6 to 7 minutes. Transfer to a plate; cover.

Transfer all but 2 tablespoons pan juices to a bowl. Add leeks to skillet over medium heat, season with salt, and cook until tender, 6 to 7 minutes. Stir in remaining 2 tablespoons pesto; cook 1 minute. Add chickpeas, 1 cup water, and reserved pan juices. Cook until slightly reduced, 6 to 8 minutes. Toss fennel with remaining 1 tablespoon oil; season with salt and pepper. Serve with chicken and chickpeas, garnished with fennel fronds.