Toast the Pine Nuts: Heat 2 tablespoons of olive oil in a 12-inch skillet or small Dutch oven over medium-low heat.
Add the pine nuts, and once the oil starts sizzling, stir occasionally until the nuts are golden brown (about 2 minutes). Transfer to a small serving bowl and set aside.
Cook the Onion, Garlic, and Chicken: In the same skillet, heat the remaining 2 tablespoons of olive oil over medium-high heat.
Add the onions with a pinch of salt and cook for about 3 minutes until lightly golden.
Stir in the garlic and cook until sizzling (about 1 minute).
Add 2 cups of shredded cooked chicken and cook for 2-3 minutes, stirring until the chicken is heated through.
Add the Beans and Stock: Stir in the drained beans and stock. Bring the mixture to a simmer, then reduce the heat to low.
Stir in the Cheese: Gradually add the grated Parmesan in batches, stirring vigorously to combine.
Once the cheese melts into the broth and the mixture turns creamy, season with salt to taste.
Turn off the heat, and while the beans are still hot, tear or chop the basil leaves and stir them in.
Serve hot in bowls. Top with the pine nuts you made earlier, add a squeeze of lemon juice, and finish with extra black pepper and grated Parmesan, if desired.