Heat 1 teaspoon olive oil in a saucepan over medium heat; cook and stir quinoa until slightly toasted and water has evaporated from quinoa, 2 to 3 minutes. Add chicken broth and bring to a boil for 5 minutes. Cover saucepan, reduce heat to low, and simmer until quinoa is tender and has absorbed the broth, about 15 minutes. Remove saucepan from heat and let sit for 5 minutes; fluff with a fork.
Mix quinoa, asparagus, chicken, tomatoes, basil, Parmesan cheese, and garlic together in a bowl. Drizzle 1/4 cup olive oil over salad and season with salt, pepper, and red pepper flakes; stir until evenly coated.