In a small bowl, combine the finely diced red onion, 2 tbsp of the lemon juice, and 1 tsp of the sea salt. Let sit for at least 20 minutes to mellow the onion’s intensity.
In a mixing bowl, stir together the pesto, ½ cup Greek yogurt, the remaining salt, and the remaining 1 tbsp lemon juice. Mix until smooth and well combined.
Lay a large flour tortilla on a flat surface. In the center, layer shredded chicken, a generous spoonful of the pesto yogurt sauce, some of the marinated red onion, arugula, and feta crumbles.
Fold the sides of the tortilla inward, then roll it up tightly like a burrito. Secure with a toothpick if needed to help keep it closed while toasting.
Heat a small amount of oil in a skillet over medium-low heat. Cook the wrap for 2–3 minutes per side, or until golden brown and lightly crisped.
Prepare the dipping sauce: In a bowl, whisk together all the dipping sauce ingredients. Add salt to taste.
Serve warm, with the lemon-garlic yogurt dip on the side.