Heat 1 tablespoon of the olive oil in a large nonstick skillet over medium-high heat and cook the onions until softened, about 3 minutes. Set aside to cool. Wipe out the skillet and reserve.
Combine the chicken, basil, parsley, oregano, ricotta, garlic, lemon zest, 1 teaspoon salt, 1/2 teaspoon pepper and the cooled onions in a large bowl. Mix well with your hands. Splash your hands with some cold water and mold the chicken mixture into 4 patties.
Combine the mayonnaise and lemon juice in a small bowl.
Heat the remaining 1 tablespoon olive oil in the reserved skillet and cook the patties until they're cooked through, 3 to 4 minutes on each side (see Cook’s Note).
Meanwhile, toast the buns in a toaster or on a dry pan and set aside.
Spread the lemon mayo on each side of the buns. Place the bottom of the bun on a plate, top with the arugula, tomato, one patty, goat cheese and finally the top of the bun. Repeat with the remaining patties.
One way to tell if the burgers are cooked through is to press your finger into the middle of a patty and if it bounces back up it's done.