Chop pistachios into teeny pieces using a food processor or a mortar and pestle. Reserve.

Slice chicken breast crosswise (against the grain) into ¼-inch-thick slices. Cut the largest slices in half. Season with salt and pepper.

Transfer pistachios to a medium-size bowl. In a second medium-size bowl, beat egg.

Pat down chicken pieces, one at a time, into pistachios, then roll in egg mixture, then pat down again into pistachios. Reserve nut-covered pieces on a plate until they are all done.

In a large frying pan, over medium/medium-high heat, add oil and butter. When butter starts to sizzle, add chicken. Cook for approximately 3 minutes on the first side, or until pistachios on the bottom are golden brown. Turn once and cook for approximately 2 more minutes, or until the second side is golden brown and the chicken is cooked throughout. 

Insert a skewer into each piece of chicken, then stick skewers in watermelon.