Heat broiler to high.
Combine rice and quinoa in a fine mesh strainer and rinse with cold water until water runs clear. Transfer to a medium saucepan add water and a pinch of salt and bring to a boil over medium-high heat.
Reduce heat to a low simmer, cover and cook undisturbed for 10 minutes. Remove from heat and let sit, covered for 10 more minutes. Fluff with a fork and set aside.
Place poblano pepper on a baking sheet and broil until skin is charred and blackened on all sides, turning frequently, about 5-8 minutes. This can also be done over a gas flame.
Put oven on bake and heat oven to 400°F. Transfer pepper to a heatproof bowl and cover with plastic wrap. Let sit until cool enough to handle, about 5-10 minutes. Once cool, remove all the blackened skin, the stem, and any seeds. Slice into thin strips.
Cut tortillas into thin strips, place on a baking sheet, toss with oil and season with salt. Spread into a single layer on the baking sheet and bake until crispy, about 10-15 minutes.
Toss rice and quinoa mixture with the 1/3 cup of Garlic and Basil dressing and divide rice among 4 bowls.
Combine chicken, cherry tomatoes, and poblano strips then divide the chicken mixture evenly between the bowls.
Divide Romaine lettuce, avocado, queso fresco, and tortilla strips between the bowls. Drizzle 1-2 tablespoon or so of dressing over everything and serve.