Cook rice according to package directions, omitting salt and fat.
Combine soy sauce and next 4 ingredients (through broth) in a small saucepan; bring to a boil. Reduce heat; simmer 3 minutes. Remove from heat.
Heat a large nonstick skillet or wok over medium-high heat. Add 2 teaspoons oil to pan; swirl to coat. Add chicken thighs to pan; cook 3 minutes on each side or until browned. Transfer to a cutting board; cool slightly. Cut into (1-inch) strips.
Return pan to medium-high heat; add remaining 1 teaspoon oil to pan. Add snow peas and onions; sauté 2 minutes. Add soy sauce mixture and chicken to pan; cook 2 minutes or until liquid is syrupy and chicken is thoroughly heated, stirring frequently. Place 1 cup rice in each of 4 shallow bowls; top each serving with 1 cup chicken mixture.