For the chile vinegar:

1. Combine the vinegar and chile in a bowl. Let stand at room temperature for at least 15 minutes.

For the stir fry:

2. Combine the chicken with 2 Tbsp water and baking soda in a bowl. Let sit at room temperature for 15 minutes. Rinse the chicken in cold water and drain well.

3. Cook noodles: Bring 6 cups of water to a boil. Place noodles in a large bowl. Pour boiling water over the noodles. Stir, then soak until noodles are almost tender, about 8 minutes, stirring once halfway through soak. Drain and rinse with cold water. Drain well and toss with 2 tsp oil.

4. Mix sauce together in a medium bowl.

5. Heat 2 tsp oil and garlic in a 12" nonstick skillet over high heat, stirring occasionally, until the garlic is deep golden brown, 1-2 minutes. Add chicken and 2 Tbsp sauce mixture, toss to coat, and spread the chicken in an even layer. Cook, without stirring, until chicken begins to brown, 1-1.5 minutes. Using tongs, flip chicken and cook, without stirring, until second side begins to brown, 1-1.5 minutes. Push chicken to one side of skillet. Add 2 tsp oil to cleared side of skillet. Add eggs to clearing. Using rubber spatula, stir eggs gently and cook until set but still wet. Stir eggs into chicken and continue to cook, breaking up large pieces of egg, until eggs are fully cooked, 30-60 seconds. Transfer chicken mixture to bowl.

6. Heat 2 tsp oil in now-empty skillet until smoking. Add broccoli and 2 Tbsp sauce and toss to coat. Cover skillet and cook for 2 minutes, stirring once halfway through cooking. Remove lid and continue to cook until broccoli is crisp and very brown in spots, 2-3 minutes, stirring once halfway through cooking. Transfer broccoli to bowl with chicken mixture.

7. Heat 2 tsp oil in now-empty skillet until smoking. Add half of noodles and 2 Tbsp sauce and toss to coat. Cook until noodles are starting to brown in spots, about 2 minutes, stirring halfway through cooking. Transfer noodles to bowl with chicken mixture. Repeat with remaining 2 tsp oil, noodles, and sauce. When second batch of noodles is cooked, add contents of bowl back to skillet and toss to combine. Cook, without stirring, until everything is warmed through, 1-1.5 minutes. Transfer to platter and serve immediately, passing chile vinegar separately.