In a large saucepan, combine the chicken broth, bouillon, rice, carrots, celery and onion. Bring to a boil. Reduce heat; cover and simmer for 20 minutes, or until rice and vegetables are tender. Stir in the chicken. Remove from heat and set aside.: As the pot heats you will notice the aroma of onion and celrots mingling with the savory steam of the chicken broth . Listen for a gentle, steady simmer rather than a roar, and watch the surface for a few soft bubbles, which mean the liquid is cooking evenly. The vegetables should become tender and slightly translucent, and the rice should swell and feel plump when you press a grain between your fingers. I always test one kernel to ensure it is soft through the center; undercooked rice will remain chalky, while overcooked rice becomes mushy and breaks down the texture of the soup. If the broth reduces too much during simmering, add a splash of warm water to keep the liquid level where it should be. After stirring in the cooked chicken , the pieces should look moist and integrate into the broth rather than separate into dry strands. A common mistake here is leaving the pan uncovered too long which can make the rice cook unevenly, so keep the lid on and check toward the end of the cook time.
In a small saucepan, melt the butter over medium heat. Whisk in the flour. Cook 1 minute, whisking constantly, until smooth and bubbly. Gradually whisk in 2 cups of the broth; cook until slightly thickened, whisking constantly. Remove from heat and set aside.: When you melt the butter , it should foam gently and smell nutty and rich. Adding the flour creates a roux, and whisking without pause ensures a satiny texture rather than lumps. You will see the mixture go from glossy to a little lighter in color, which indicates the raw flour is cooking off. Gradually introducing warm broth warms the mixture evenly, helping it thicken into a velvety consistency that will cling to the back of a spoon. If you see any tiny bits of flour or graininess, keep whisking over low heat until completely smooth. A frequent error is rushing the addition of cold liquid which can shock the roux and encourage lumps, so take your time and whisk steadily.
In a small bowl, beat the eggs until frothy. Gradually beat in lemon juice and 2 cups of the thickened broth mixture. Slowly add the egg mixture to the rest of the broth mixture in the large saucepan, stirring constantly. Heat gently until the soup thickens enough to coat a spoon, stirring frequently (do not boil). Add salt and pepper to taste.: The eggs should become pale and frothy when beaten, and whisking in the lemon juice brightens the mixture and tempers the eggs. This tempering step is crucial because adding the warm thickened broth slowly to the eggs keeps them from scrambling. As you combine the tempered egg lemon liquid back into the main pot, stir in a steady circular motion so the proteins disperse evenly and create a silky texture. Heat the soup gently; you will notice it slightly thicken and leave a thin film on a spoon, signaling the right moment to stop. Overheating or boiling will cause the eggs to coagulate into curds, producing an undesirable texture. If small curdles appear, immediately remove from heat and whisk vigorously off the heat to smooth them out. Taste and adjust seasoning with salt and pepper sparingly because the bouillon may already add sodium.
Ladle into soup bowls and garnish each with a lemon slice and a sprinkle of green onions.: The final act is visually and aromatically important. As you ladle, watch the way the broth coats the spoon, the way tender chicken pieces rest among the plump rice , and how the surface catches light. A thin lemon slice floating on top announces the citrus note, and the fresh scent of sliced green onions wafting up enhances the first mouthful. Serve immediately so the garnishes remain crisp and the soup retains its gentle steam. A common oversight is letting the soup sit too long before serving which dulls the bright lemon aroma and softens the green onion crispness, so time your plating so bowls reach the table right away.