Sprinkle both sides of each chicken thigh fillet with salt and pepper. Place in a resealable bag, add the flour, seal the bag and shake to coat each fillet well with flour.
Heat enough cooking oil in a large frying pan to reach a depth of one inch.
Lay the chicken fillets in the hot oil in a single layer and cook at 350F for three to four minutes per side depending on the thickness of the fillets. Scoop out the fried chicken fillets and move to a chopping board.
In a small sauce pan, mix together the lime juice, lemon juice, chicken broth and about two tablespoons sugar. Taste. Add fish sauce to balance the saltiness and more sugar if the mixture is still too tart. Keep adjusting until you get the balance that you like. Stir the starch into the sauce.
Cook the sauce over medium heat, stirring often, until thickened and clear. Lower the heat and cook, covered, for another two to three minutes to get rid of any starchy taste.
Off the heat, drop the cubes of butter into the hot sauce. Stir gently until the butter is melted and fully incorporated into the sauce.
Cut the fried chicken fillets into strips. Arrange on a serving platter. Garnish with lime and lemon slices (those who find the sauce still on the sweet side and simply squeeze the lime and lemon juices into the sauce on their plates). Pour in the sauce.
Serve the lemon lime butter chicken at once.