How to Make It
Place quartered onion and ginger, cut sides down, in a stockpot, and add garlic cloves. Cook over medium-high until vegetables are deeply charred but not burned, about 4 minutes per side. Add oil and shrimp shells, and cook, stirring, until shells are bright pink, about 3 minutes. Add wine, and stir to combine, scraping browned bits from bottom of pot. Bring to a boil, and simmer until reduced by half, about 2 minutes. Add 6 cups water, and bring to a simmer. Cook until onions are tender, about 10 minutes. Pour mixture through a fine wire-mesh strainer into a bowl; discard solids. Set aside stock.
Heat a large Dutch oven over medium. Add bacon and cook, stirring often, until crispy and fat has rendered, about 8 minutes. Using a slotted spoon, transfer bacon to a paper towel–lined plate.
Add minced onion to drippings in Dutch oven and cook over medium, stirring often, until softened, about 5 minutes. Add carrots and sweet potato, and cook, stirring occasionally, 6 minutes. Add turnips and cook, stirring occasionally, until vegetables are soft around edges, 2 to 3 minutes. Stir in tomato paste and cook until brick red, 1 to 2 minutes. Add stock and bring to a simmer.
Add clams to Dutch oven, ensuring they are fully submerged, and add shrimp. Cover and simmer 3 minutes. Add mussels; cover and simmer until clams and mussels have opened and shrimp are opaque, 2 to 4 minutes. (Discard any mussels that do not open.) Taste broth and add salt, if needed. Serve stew topped with dollops of aioli and crispy bacon.