Preheat the oven to 400°F. Grease an 8×8-inch or 9×9-inch casserole dish with a little of the 1 tablespoon extra-virgin olive oil; set the dish aside and reserve the rest of the oil for cooking.
Fill a medium pot and a large pot with enough water to cover the chicken and the cauliflower florets, respectively. Bring both pots to a boil, then reduce heat to medium.
Add the 8 ounces chicken breast to the medium pot and the 1½ pounds cauliflower florets to the large pot. Cover only the cauliflower pot. Cook until the chicken is no longer pink in the center and the cauliflower is fork-tender, about 10–15 minutes.
When the chicken is done, transfer it to a plate to cool. Drain the cauliflower in a colander and transfer it to a clean kitchen towel; let it cool for about 10 minutes or until cool enough to handle.
While the chicken and cauliflower cook, heat the remaining olive oil in a large, high-sided pan over medium heat. Add the 1 tablespoon minced garlic, the roughly chopped large onion, and the 1 tablespoon taco seasoning. Cook, stirring frequently, until the onion is softened and lightly golden, about 3–5 minutes.
Stir in the entire 10-ounce can red enchilada sauce. Bring the mixture to a boil, cook at a full boil for 1 minute, then reduce heat to medium and simmer until the sauce reduces and thickens slightly, about 7 minutes. Remove from heat.
Once the cauliflower is cool enough to handle, wrap it in the kitchen towel and wring out as much moisture as possible. Repeat once using a clean towel.
Place the drained cauliflower in a large food processor. Add the 1 large egg, 1 egg white, ½ teaspoon salt, and black pepper to taste. Pour in the cooked enchilada sauce mixture. Process until smooth, then transfer the blended mixture to a large bowl.
Shred the cooled chicken and add it to the bowl with the cauliflower-sauce mixture. Add the 4-ounce can diced green chilis and stir until evenly combined.
Transfer the mixture to the prepared casserole dish and spread it evenly. Cover the dish tightly with foil and bake for 30 minutes.
Remove the foil and continue baking until the casserole feels set and the edges are lightly browned, 30–40 minutes more.
Sprinkle the top with 1 cup reduced-fat Mexican cheese blend and return to the oven just until the cheese melts, about 2–3 minutes.
Remove from the oven, tent with foil to keep warm, and let the casserole rest for 10 minutes to absorb excess moisture. Slice and serve.
Garnish servings with cilantro, diced tomatoes, and sliced avocados. Serve hot.