Combine 2 teaspoons vinegar, miso, and 5 teaspoons water in a small bowl, stirring with a whisk.
Combine remaining 2 tablespoons vinegar, 2 teaspoons oil, soy sauce, and sugar in a large bowl. Add kale, radishes, and green onions to bowl; toss to coat.
Heat a large nonstick skillet over medium-high heat. Add butter and remaining 2 teaspoons oil to pan; swirl to coat. Add garlic; sauté 30 seconds. Add rice; sauté 5 minutes or until lightly browned. Stir in kale mixture; remove pan from heat. Remove kale mixture from pan; keep warm. Wipe pan with paper towels.
Return pan to medium-high heat. Gently break eggs into pan. Pour 1/4 cup water around eggs; cover and cook 3 minutes or until desired degree of doneness. Place about 1 1/4 cups kale mixture in each of 4 shallow bowls. Top each serving with 1 egg and about 1 tablespoon miso mixture. Sprinkle evenly with sesame seeds.
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This was delicious! I was about a 3/4 cups short on radish, so I used julienned carrots to make up the difference. I enjoyed the additional color and texture and will add the carrots again the next time I make it.