In a large bowl, whisk together all dressing ingredients and season to taste with salt and pepper.
Add chickpeas, fennel, and brussel sprouts to bowl with dressing and toss to combine.
In a large skillet over medium heat, heat 1 tablespoon oil. Add halloumi and cook until charred and golden on both sides, about 4 to 6 minutes total. Remove cheese to a large plate.
In the same skillet over medium heat, heat remaining oil. Add pita, sprinkle with sumac and toast until golden and crisp, about 4 minutes. Remove to plate with halloumi.
Add arugula, halloumi, and pita to bowl with chickpeas, fennel, and sprouts, toss to combine. Add orange segments and avocado slices and toss very gently until combined. Season to taste with salt, pepper, and a dash red pepper flakes if desired, and serve.