Measure the water and the brown rice into a pot. Cover and bring to a boil over high heat. Reduce the heat, cover, and cook at a simmer for about 20 minutes, until the rice is tender but the water is not completely absorbed. Move the covered pot of rice to a cool burner on the stove for later in the recipe.
Melt the butter in a small skillet over medium heat. Add the beaten eggs. Sprinkle lightly with sea salt and alittle white pepper. Cook for about 2 minutes, until the edges are just set. Mix the eggs with a fork to break them up into big pieces, and continue cooking and mixing for about a minute until the eggs are cooked but not dry. Set aside.
Heat the olive oil in a large skillet over medium heat. Add the onion, garlic, shallots, and ginger. Sprinkle lightly with sea salt and white pepper. Stir to combine. Cook for about 5 minutes, stirring occasionally until the shallots are golden and the onions have softened. Add the carrots, broccoli, and mushrooms. Use two wooden spoons to toss and mix. Continue cooking for about 4 minutes, occasionally tossing the vegetables until the carrots are crisp-tender. Stir in the frozen peas and the tamari. Cook for 1-2 minutes, tossing occasionally until the peas are heated through. Add the cooked rice, using a rubber spatula to scrape the pot of the rice and any liquid. Stir well to combine. Add the scrambled eggs and stir to combine. Taste for seasonings.