Make the tzatziki first so flavors can mellow: grate the cucumber into a bowl, sprinkle with a pinch of salt and press or squeeze out as much moisture as possible using a clean towel or paper towels. Transfer to a medium bowl and stir in the Greek yogurt, minced garlic, 1 tablespoon lemon juice, chopped dill, 1 tablespoon olive oil, 1/2 teaspoon salt and 1/4 teaspoon black pepper. Cover and refrigerate for at least 10 minutes.
Start the rice: combine the rinsed rice and 2 cups water in a small saucepan and bring to a simmer over medium-high heat. Reduce to low, cover and cook until the water is absorbed and rice is tender, about 15–18 minutes. Remove from heat and let sit covered for 5 minutes, then fluff with a fork.
While the rice cooks, flatten the chicken to even thickness (about 3/4 inch) for quick, even cooking. In a bowl, whisk 1 tablespoon olive oil, 1 tablespoon lemon juice, 2 minced garlic cloves, oregano, 1/2 teaspoon salt and 1/4 teaspoon pepper. Add the chicken, coat well and let marinate at room temperature for 10–15 minutes.
Heat a large skillet over medium-high heat and add 1 tablespoon olive oil. When shimmering, add the chicken and cook until golden and an instant-read thermometer registers 165°F, about 5–7 minutes per side depending on thickness. Transfer to a cutting board and let rest 5 minutes before slicing thinly.
As the chicken rests, assemble the salad mix: divide the chopped romaine (or greens), halved cherry tomatoes, red onion and olives among four bowls. Spoon warm rice into each bowl alongside the greens.
Slice the rested chicken across the grain into strips and distribute over the bowls. Spoon generous dollops of chilled tzatziki over each portion of chicken and rice.
Finish each bowl with crumbled feta, a light drizzle of olive oil and a sprinkle of ground sumac for a bright, tangy lift. Taste and adjust salt and pepper if needed, then serve immediately.