Preheat oven to 375º. In a large pot of salted boiling water, cook lasagna noodles until very al dente. Drain.
Meanwhile, in a large skillet over medium heat, heat oil. Season chicken with oregano, salt, and pepper. Cook in a single layer, in batches if necessary, until golden and cooked through, 8 minutes. Return all chicken to skillet, then add balsamic vinegar and half the garlic to pan and stir until balsamic is slightly thickened, 1 minute. Transfer to a plate.
Make sauce: Bring skillet back to medium heat and melt in butter. Add remaining garlic and cook until fragrant, 1 minute. Add flour and cook until golden, 1 minute more. Gradually add in milk while whisking, then simmer until thick and creamy, 3 minutes. Add Parmesan and 1 cup mozzarella, stirring until melty. Season with salt and pepper.
In a small bowl, stir together ricotta and egg and season with salt and pepper.
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 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 for a total of three layers, ending with mozzarella.
Bake until bubbly and cheese is melty, about 25 minutes. Turn oven to broil and broil until lightly golden, 2 to 3 minutes.
Garnish with basil and drizzle with balsamic glaze before serving.