Make the kombu stock: In a small bowl, combine the kombu and 3 1⁄2 cups water; chill at least 8 hours or overnight.
Start the sauces: In a small bowl, add the almonds and enough cool water to cover. In a separate small bowl, add 1⁄2 cup pistachios and enough water to cover; let the nuts soak at least 8 hours or overnight.
In a small pot over low heat, add the kombu mixture; heat to just below a simmer (about 150°), then continue cooking at that temperature, adjusting the heat as needed, for 1 hour. Remove and let cool.
Using a slotted spoon, remove the kombu and squeeze any moisture back into the pot; coarsely chop the greens and reserve. Chill the kombu and stock until ready to use, up to 1 week.
Make the almond cream: In a blender, add the drained almonds and 1 1⁄3 cups of the chilled kombu stock (reserve the remaining stock). Blend the mixture into a thick, smooth cream, about 1 minute. Strain through a fine sieve set over a bowl; discard the solids.
Make the relish: In a large pan over medium-high heat, heat 2 tablespoons olive oil. Add the parsley, scallions, and a pinch of salt and cook, stirring occasionally, until the greens wilt and darken, 1–2 minutes. Transfer to a plate and spread into an even layer; chill until cooled.
In the clean bowl of a blender, add the chilled greens, soaked and drained pistachios, and remaining 1⁄2 cup olive oil. Blend until pestolike and almost smooth, about 2 minutes.
Transfer the herb mixture to a medium bowl. Finely chop the remaining 3 tablespoons pistachios. Add them to the bowl along with 1⁄4 cup of the reserved chopped kombu, 2 tablespoons of stock, the chile paste, lemon peel, soy sauce, and Worcestershire sauce; stir to combine.
Prepare the vegetables: Using a mandoline or sharp knife, thinly slice a few carrots, radishes, and/or cucumbers to serve raw, as garnish. Pair like vegetables together so they can be cooked in groups. Peel carrots, turnips, and beets, leaving a bit of the stems attached for appearance; halve or slice any large ones vertically so they are about the same thickness. Trim radishes and spring onions. Cut them, as well as cucumbers, zucchini, and romano beans, into bite-size pieces. Shell fava beans and peas. Stem any tomatoes and cut in half.
In a 4-quart Dutch oven or heavy-bottomed pot with a lid, heat 2 tablespoons olive oil over medium-high heat. Add a generous pinch of salt and the first batch of vegetables, stirring to coat evenly with the oil; cover and let cook until the vegetables begin to soften, 5–7 minutes. Stir in 2 tablespoons of stock (or up to 3 for firmer vegetables) and another small pinch of salt; cover and cook until the liquid has evaporated and the vegetables are al dente and glazed, 2–3 minutes more. Repeat with the remaining groups of vegetables, adding up to 1 tablespoon olive oil more for each batch, if necessary.
When ready to serve, spoon some of the almond cream onto 6 individual plates or one large serving platter. Spoon small dollops of the herb relish around the plate. Drizzle with a few drops of olive oil and sprinkle with seaweed flakes if using. Arrange the cooked and raw vegetables atop the sauce and serve with the remaining almond cream and herb relish on the side as desired.