Heat a large non-stick skillet or wok over medium-high heat with half the sesame oil.
Season shrimp with salt and pepper, then sear until pink and opaque, about 2-3 minutes; remove and set aside.
In the same pan, add the remaining sesame oil, garlic, and the white parts of the scallions, sautéing until fragrant.
Stir in the chopped kimchi and gochujang, cooking for 2 minutes to caramelize the sugars.
Add the chilled brown rice and tamari, pressing down with a spatula to break up clumps and let the rice crisp slightly.
Return the shrimp to the pan and toss everything together until heated through.
In a separate small non-stick pan, fry the eggs to your preference (sunny-side up or over-easy).
Divide the rice into bowls, top with the fried eggs, and garnish with the green parts of the scallions.