Add the rice and water (1.5 cups water for 3/4 cup rice) in a medium pot, and add a big pinch of salt. Heat on high until boiling,stir, then cover and decrease temperature to low. Simmer for 20 minutes, then take off heat without removing lid.
While the rice cooks, prepare the veggies. Cut off and discard the bottom 1/2 inch of the broccoli stem; cut the broccoli into small florets. Cut off and discard the stems of the peppers; remove the cores, then thinly slice lengthwise. Grate the garlic with a microplane. Thinly slice the scallions, separating the white bottoms and the green tops. In a medium bowl, scramble the eggs.
Heat half the sesame oil on medium-high until hot. Add the broccoli florets and cook, without stirring, 2 to 3 minutes, or until lightly browned. Add 2 tablespoons of water and season with salt and pepper. Cook, stirring occasionally, 2 to 3 minutes, or until the broccoli is slightly softened and the water has cooked off. Add the sliced peppers, garlic, sliced white bottoms of the scallions, and as much of the red pepper flakes as you’d like. Season with salt and pepper. Cook, stirring occasionally, 2 to 3 minutes, or until softened. Transfer to a large bowl. Wipe out the pan.
In the same pan, heat the remaining sesame oil on medium-high until hot. Add the cooked rice in an even layer. Cook, without stirring, 3 to 4 minutes, or until slightly crispy. Add the soy sauce and vinegar. Cook, stirring frequently, 30 seconds to combine.
Push the rice to one side of the pan and pour the eggs onto the pan and stir continuously to scramble. Mix the eggs into the rice. Toss the veggies back into the pan and mix to combine. Taste once more and add a bit more soy sauce if needed.