Tip:The burgers can be shaped up to 2 hours ahead and refrigerated.
Put the egg yolk, capers, garlic, lemon juice, salt, and pepper in a food processor and purée until smooth. With the motor running, slowly drizzle the oil through the feed tube to form an emulsion. Stop the motor, add the 2 Tbs. chopped mint, and pulse to combine. Spoon 1/4 cup of the aïoli into a small bowl.
Add the tuna and egg white to the food processor bowl and pulse until just chopped. Line a baking sheet with waxed paper or parchment. Drop 18 rounds of the tuna mixture by heaping tablespoons onto the baking sheet. Use your hands to shape the mounds into mini burgers about 1/3 inch thick.
Heat a gas grill to high or prepare a hot charcoal fire.
Brush one side of the tuna burgers with oil and season with salt and pepper. Gently flip the burgers and oil and season the other side. Put the pita triangles on another baking sheet, brush both sides with oil and sprinkle both sides with a little salt and pepper. Grill the pitas on both sides until lightly browned and a little crisp but still pliable, 1 to 2 minutes total. Transfer to a serving platter.
Grill the burgers (covered on a gas grill, uncovered on a charcoal grill) on one side until they have nice grill marks, about 2 minutes. Flip the burgers, loosening them with a metal spatula if necessary, and grill the other side until marked and just cooked through, about 2 minutes more.
While the burgers are cooking, spread a little of the reserved aïoli inside each pita. As the burgers come off the grill, tuck one into each pita along with a mint leaf and serve immediately.