Step 1 In a nonreactive, shallow baking dish, stir together 2 tablespoons of the olive oil, lemon zest and crushed garlic. Add the chicken breasts and rub them all over with the marinade. Cover loosely with plastic wrap and let stand at room temperature for 30 minutes.
Step 2 While the chicken is marinating, make the pesto. Puree the watercress, minced garlic, almonds and one-fourth teaspoon salt in a food processor until smooth. With the motor running, drizzle in the remaining one-third cup olive oil until emulsified. Set aside.
Step 3Heat a grill pan or grill over medium-high heat. Shake off the excess marinade and season each chicken breast with one-fourth teaspoon salt and a couple grinds of black pepper. Place the chicken on a medium-hot grill and cook until the meat is opaque and firm and a thermometer inserted in the center reads 160 degrees, 10 to 12 minutes. Cool, then cut into a one-fourth-inch dice.
Step 4Heat the oven to 375 degrees. Lightly brush one sheet of filo dough with melted butter. Top with a second sheet and brush with butter. Top with a third sheet. Use a sharp knife to cut the filo dough crosswise into eight strips. Spoon about 1 teaspoon diced chicken at one end of the filo strip. Top with one-half teaspoon of the pesto, then about one-fourth teaspoon cheese.
Step 5Fold one corner over the filling, forming a small triangle. Continue to fold, as if folding a flag, retaining the triangular shape. The last fold may need trimming. Repeat with the remaining filling and filo strips. Place turnovers in a single layer on parchment-lined baking sheets, lightly brush with melted butter. Bake till golden brown, 15 to 20 minutes.