Add the olive oil and 1/4 cup kumquats to a large non-stick skillet over low heat.
Season the chicken with salt and pepper on both sides.
Heat the olive oil to 200 degrees and add the chicken to the skillet. Cook for 10 minutes, then flip the chicken and continue cooking for an additional 10-15 minutes or until cooked through.
Transfer the chicken to a platter and keep warm.
Add the radishes to skillet and cook for 5-7 minutes or until tender. Using a slotted spoon, transfer the radishes to the platter with the chicken and keep warm.
Add the remaining kumquats, olive oil, basil leaves and pine nuts to a food processor. Process until smooth.
Spoon the citrus basil over the chicken and serve immediately.