In a small bowl, whisk together brown sugar, soy sauce, lime juice, 1/2 teaspoon red curry paste and 1/8 teaspoon salt. In a separate small bowl, whisk together eggs and remaining red curry paste.

Chop green onions on the diagonal; chop white parts into 1-inch pieces and set aside separately.

Heat a large wok or nonstick skillet over medium-high heat. Add 2 teaspoons oil; swirl to coat pan. Add rice and stir 1 1/2 minutes, then transfer to a large bowl. Add another 2 teaspoons oil to the pan. Add shallots and cook 30 seconds, stirring constantly until soft. Add shrimp and cook 2 minutes, until shrimp is heated through.

Add brown sugar-soy sauce mixture to pan. Bring to a boil, then reduce heat to a simmer. Simmer 2 minutes until liquid thickens and reduces slightly. Add contents of pan to rice in large bowl.

Return pan to medium-high heat. Add another 2 teaspoons oil, then add broccoli and red bell pepper. Stir 3 minutes until just tender. Add remaining 1/8 teaspoon salt, white parts of green onions and minced garlic. Stir 1 minute until fragrant. Return shrimp and rice mixture to pan and stir ingredients until combined and heated through. Transfer all contents back to large bowl.

Return pan to medium-high heat. Add remaining teaspoon oil; swirl to coat pan. Add egg mixture and cook 1 minute until just set. Transfer egg to a cutting board and roughly chop. Stir into shrimp-rice mixture along with half of the remaining chopped green onions. Serve with more green onions on top and lime wedges on the side.