This makes a light breakfast or an appetizer. If you want a more substantial breakfast, double the recipe and have two of these silky smooth savory Japanese custards. You can also split this between 4 ramekins for a light appetizer.
Cover the shiitake with boiling water to rehydrate them and set aside.
In a bowl, whisk the eggs with the rice vinegar, monk fruit, and soy sauce followed by the dashi. Strain the mixture through a fine mesh strainer to remove as many bubbles and any non-homogenous egg bits (to keep the custard smooth).
Chop the shiitake and shrimp into bite sized pieces and divide them among two ramekins. Top with carrots, if using then pour the egg mixture evenly between the two. Set into a glass baking dish and place into the oven. Pour boiling water about halfway up the side of the custard dishes and bake until the custards are set but wobble in the center when shaken gently, 30 minutes.
Remove the custards from the oven and water bath and let cool slightly before serving.