Warm the sesame oil in a medium pot over medium heat. When the oil loosens and looks glossy across the bottom of the pot, it is ready for the aromatics.
Add the diced onion and cook for 2 minutes. The pieces should begin to soften at the edges while staying pale and fragrant.
Stir in the minced garlic and shredded bok choy, then cook for 1 minute. The garlic should smell nutty and the bok choy should start to relax without browning.
Pour in the vegetable broth or daishi stock and bring the mixture to a gentle simmer. Look for small, steady bubbles around the edges instead of a hard rolling boil.
Loosen the white miso paste with some warm broth, then stir it into the pot until smooth. The broth should take on a soft, cloudy swirl with no dense pockets of paste.
Keep the soup at a gentle simmer for 10 minutes. The bok choy stems should become tender while the broth stays aromatic and calm.
While the soup simmers, press the block of tofu with paper towel and cut it into small cubes. The cubes should feel firm and dry on the outside rather than wet or fragile.
Slide the tofu into the soup and stir gently until warmed through. The cubes should remain intact and feel warm and springy when lifted with a spoon.
Finish the soup with a squeeze of fresh lemon juice. The citrus should brighten the aroma without making the broth taste sour.
Top with sliced green onions if desired, then serve. The fresh green garnish adds a crisp finish against the silky, savory broth.