Prepare the broth and mushrooms. Gently warm the dashi in a saucepan (do not let it boil). If using dried shiitakes, soak them in the warm dashi for about 10 minutes until tender.
Remove the mushrooms and thinly slice all the mushroom caps.
Sauté the vegetables.
In a deep skillet or wide saucepan with a lid, heat the oil over medium-high heat.
Add the chopped onion, sliced mushroom caps, burdock root (if using), and carrot.
Cook, stirring occasionally, for about 10 minutes until the vegetables soften and the mushroom edges begin to brown.
Toast the rice.
Reduce the heat to medium, add the rinsed rice to the skillet, and stir for one minute until the grains are well-coated with oil and combined with the vegetables.
Simmer the rice.
Stir in the peas, warm dashi, soy sauce, and mirin (or honey).
Bring the mixture to a gentle simmer, then reduce the heat to medium-low, cover the pan securely, and cook for 15 minutes.
Finish and rest. Uncover the pan. The rice should be mostly cooked but still slightly soupy. Increase the heat slightly and cook for another 2-3 minutes, stirring gently, until the rice is tender and the liquid is absorbed, but the dish remains moist.
Serve.
Taste and season with additional salt or soy sauce if needed.
Serve immediately, garnished with crumbled toasted nori on top.
Don't discard your shiitake stems! Save them to add a flavor boost to your next batch of vegetable stock.