Heat oil in a Dutch oven over medium-high. Add carrot and onion; cook, stirring often, until vegetables are softened, 3 to 4 minutes. Add garlic and crushed red pepper; cook, stirring constantly, until fragrant, about 1 minute. Add stock to Dutch oven; increase heat to high, and bring stock to a boil. Add orzo, and cook, uncovered, until orzo is al dente, about 6 minutes. Meanwhile, whisk together eggs and lemon juice in a medium bowl until frothy. Once orzo has cooked 6 minutes, carefully remove 1 cup boiling stock from Dutch oven. Gradually add hot stock to egg–lemon juice mixture, whisking constantly to temper the eggs, about 1 minute. Pour egg mixture back into Dutch oven, and stir to combine. Reduce heat to medium-low; stir in chicken, spinach, salt, and pepper. Cook, stirring constantly, until spinach wilts, about 1 minute. Divide soup among 6 bowls; sprinkle servings evenly with dill.