Prepare all ingredients as listed above.
Place cucumber slices in a pitcher with water and refrigerate to infuse. Serve over ice if desired.
Rinse brown rice under cold water and cook according to package directions until tender. Season lightly with salt and pepper and keep warm.
Toss sliced cucumber with salt and let sit for 5 minutes. Gently squeeze out excess liquid.
Add rice wine vinegar, sesame oil, gochugaru, sesame seeds, honey, scallions, and sliced garlic to the cucumbers. Toss well and set aside.
Heat vegetable broth gently in a small saucepan over low heat until warm. Do not boil. Keep warm until serving.
Heat half of the sesame oil in a large pan over medium heat. Add minced garlic and sauté about 30 seconds until fragrant.
Add carrots, zucchini, and bean sprouts. Stir-fry for 4–6 minutes until tender but still vibrant.
Stir in soy sauce and sliced scallions, then remove vegetables from heat.
Heat the remaining sesame oil in a separate pan over medium heat. Fry eggs sunny-side up or over-easy until the whites are set.
Portion brown rice into bowls.
Serve the sautéed vegetables over the rice along with a fried egg, kimchi, cucumber banchan, a spoonful of gochujang if desired, and sesame seeds. Serve with warm vegetable broth and cucumber water.