Cook rice: Rinse rice in a fine-mesh sieve until water runs clear. Combine rice, 1¼ cups water, and ½ teaspoon salt in a small saucepan and bring to a boil. Reduce heat to low, cover, and cook until rice is tender and water is absorbed, about 17 minutes. Remove from heat. Keep covered until ready to serve.
Prep ingredients: Peel and finely chop 1 teaspoon garlic. Peel half of the ginger, then finely chop (save rest for your own use). Trim scallions; cut into 1-inch pieces. Using a vegetable peeler, remove 4 (1-inch) wide strips of orange zest. Squeeze ½ cup orange juice into a small bowl. Trim ends from broccoli; cut crown into 1-inch florets. Pat chicken dry and cut into 1-inch pieces.
Make sauce: Whisk apricot preserves, all of the tamari, and 1 tablespoon sugar into the bowl with orange juice until sugar is dissolved.
Stir-fry vegetables: Heat 1 tablespoon oil in a medium skillet over medium-high. Add broccoli and a pinch each of salt and pepper. Cook until broccoli is just tender and browned in spots, 3–4 minutes. Add scallions and orange zest to skillet, and cook until browned in spots, about 1 minute. Transfer veggies to a bowl.
Brown chicken: Heat 1 tablespoon oil in same skillet over high. Toss chicken with salt, pepper, and 1½ tablespoons flour. Add chicken to skillet and cook, stirring occasionally, until browned, 3–4 minutes minutes. Add chopped garlic and ginger, orange zest, and 2 teaspoons oil; cook, stirring, until aromatics are fragrant, about 1 minute.
Finish & serve: Add sauce to skillet. Bring to a boil, and cook, stirring, until sauce is thickened slightly, 1–2 minutes. Stir in broccoli and scallions; cook until warm, about 1 minute. Season to taste with salt and pepper. Fluff rice with a fork. Serve orange chicken stir-fry over rice.