Let's cook.
In a medium bowl, gently whisk the eggs until just combined, being careful not to create too many bubbles. Do not overbeat.
In a separate bowl, combine the dashi broth, soy sauce, mirin, and salt. Stir until the salt is dissolved.
Slowly pour the dashi mixture into the whisked eggs, stirring gently to combine. Strain the mixture through a fine-mesh sieve into a clean bowl to remove any egg white strands or bubbles, ensuring a smooth custard.
Divide the chicken pieces, sliced shiitake mushrooms, and shrimp evenly between two heatproof chawanmushi cups or small ramekins.
Carefully pour the strained egg mixture over the ingredients in each cup, filling them about 3/4 full.
Prepare a steamer. Place the cups in the steamer basket, ensuring they are not touching each other. Cover the cups loosely with foil or their lids if they have them, then cover the steamer with its lid.
Steam over medium-low heat for 15-20 minutes, or until the custard is just set but still slightly wobbly in the center. A skewer inserted into the center should come out clean.
Carefully remove the cups from the steamer. Serve immediately while warm.
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