LASAGNA: In a large pot of salted boiling water, cook lasagna noodles until al dente; drain.
CHICKEN: Meanwhile, in a large skillet over medium heat, heat oil; season chicken with oregano, salt, and pepper. Cook until golden and cooked through, 8 minutes. Add balsamic vinegar and half the garlic to pan and stir until combined. Transfer to a plate.
WHITE SAUCE: In the same skillet, melt butter over medium heat. Add remaining half garlic and cook until fragrant, about 1 minute. Add flour and cook until golden, about 1 minute more. Add milk and simmer until thick and creamy, about 3 minutes, then add 1 cup mozzarella and Parmesan; season with salt and pepper.
CHEESE FILLING: In a small bowl, stir together ricotta and egg and season with salt and pepper.
TO ASSEMBLE: In a large baking dish, spread a thin layer of sauce. Add a layer of lasagna noodles, slightly overlapping, and top with about a third of ricotta cheese mixture, a third of tomatoes, a third of the basil, a third of the cooked chicken, another layer of sauce, and a third of mozzarella. Repeat with three layers, ending with mozzarella.
Bake until bubbly and golden, 35 minutes. Garnish with basil and drizzle with balsamic glaze before serving.
BALSAMIC GLAZE: Pour 1/2 cup balsamic vinegar in a small saucepan over medium high heat. Simmer until reduced by half.