In a medium saucepan, bring 4 tablespoons salt and 4 cups water to a boil, stirring to dissolve the salt. Remove the pan from the heat and let cool. Pour the salt water into a bowl, stir in the sunchokes, and refrigerate at least 8 hours or overnight. The next day, drain the sunchokes and return to the bowl. In a medium saucepan, toast the celery seeds and peppercorns over medium-high heat until fragrant, about 1 minute. Pour in 2 tablespoons salt along with both vinegars, the sugar, Aleppo pepper, garlic, and bay leaf and bring to a boil. Pour the brine over the sunchokes, let cool completely, and then cover and refrigerate at least 8 hours or up to 2 weeks.
In an 8-qt. saucepan, heat the olive oil over medium-high. Add the carrots to the oil, cut sides down, and cook, without turning, until dark golden brown, 3 to 4 minutes. Using a slotted spoon, transfer the carrots to a plate. Repeat cooking the red radishes, kohlrabi, parsnips, turnips, and daikon in individual batches until all the vegetables are browned. Add the garlic halves, cut sides down, to the oil and cook until golden brown, about 2 minutes. Flip the garlic halves and return all the browned vegetables to the pan along with the thyme and 2 cups water. Season with salt and bring the water to a boil. Reduce the heat to maintain a steady simmer and cook, covered, until the vegetables are tender, about 20 minutes. Using a slotted spoon, lift the vegetables from the pan and transfer to a large plate; discard the garlic halves and thyme. Return the pan to the heat, bring to a boil, and cook until the liquid reduces to 2 tablespoons, about 12 minutes. Pour the liquid into a small bowl and let cool.
In a small saucepan, combine the quinoa with 1 cup water and season with salt. Bring to a boil, reduce the heat to medium-low, and cook the quinoa, covered, until tender, 12 to 15 minutes. Scrape the cooked quinoa into a bowl and let cool.
In a small skillet, brown the butter over medium heat. Pour the butter through a fine sieve into the bowl with the reduced cooking liquid. Stir in the fish sauce and season the dressing with pepper. Pour half the dressing over the quinoa along with the lemon zest and juice and toss to combine. Transfer the quinoa to a serving platter and arrange the vegetables over the top. Drizzle with the remaining dressing and top with the sunchokes before serving.