Prep the rice. Put the rice in a medium-sized bowl with enough warm water to completely cover. Mix the rice around with your hands then drain off the water. Repeat three to four more times with fresh water, or until the water is no longer cloudy.
Place the prepped rice in a medium-large pot with the vegetable broth and 1 1/2 cups water. Cover with a lid and bring to a low boil. Reduce the heat and gently simmer for 1 hour, leaving the lid on the pot. Stir once or twice during the cooking process.
Remove the rice from heat and crack the lid, then let the rice sit for 15 minutes, allowing some more moisture to evaporate, and the congee to thicken.
While the rice is resting, heat a medium-large skillet over medium heat. Add the oil, swirl to coat, then add the mushrooms in a single layer. Let the mushrooms cook, undisturbed, for 5 minutes, then stir and continue cooking until softened, about 3 minutes more. Set aside.
Remove the congee lid, then bring the rice back to a simmer.
Add the ginger, red pepper flakes, and frozen peas. Cook for 1 to 2 minutes, stirring occasionally, until the peas are hot.
Turn off the heat, then stir in the liquid aminos (or tamari) and miso. Stir until everything is well-incorporated. Taste, and adjust if you'd like.
Spoon the congee into bowls, then top with the mushrooms, toasted sesame oil, scallions, and sesame seeds or sesame salt.