Cook the pasta in a large pot of well salted boiling water for 1 minute less than al dente, according to package instructions. During the final 2 minutes of the cooking time, add the broccoli florets to the boiling water with the pasta. Drain.
Meanwhile, melt the butter in a large skillet over medium heat. Add the garlic and stir until fragrant, about 30 seconds. Add the chicken broth, increase the heat, and bring the mixture to a boil. Reduce the heat so that the liquid is at a gentle simmer. Stir in the cream, pesto, and Parmesan until smooth.
Add the cooked pasta, broccoli, and chicken to the sauce. Let it bubble and thicken slightly, stirring frequently, until all of the ingredients are warmed through, about 2 more minutes. Taste and season with salt and pepper, if necessary. Garnish with toasted pecans and other optional toppings if desired.