Make the vinaigrette:
In a large container, combine 2 tablespoons water with the soy sauce, fish sauce, vinegar, sesame oil, red pepper flakes, brown sugar, shallot, garlic, ginger, and perilla leaves. Mix well and make sure the leaves are fully submerged. Let them marinate at room temperature for at least 1 hour or as long as 12 hours.
Put the mixture into a blender and pulse until the ingredients are blended but the sauce still has a lot of texture. Transfer the mixture to a medium bowl and whisk in the olive oil. Adjust the seasoning with additional soy sauce and salt, if desired.
Make the Mussels:
In a 3- to 4- quart Dutch oven or heavy-bottomed pot, heat the olive oil over medium until simmering. Add the shallots and fennel, and season with salt. Sauté until both are translucent, 5 to 7 minutes. Add the garlic and thyme, and cook until fragrant and wilted, about 1 minute. Pour in the wine and bring to a boil.
Add the vinaigrette, mussels, and butter, stirring once or twice so that the aromatics are well distributed. Cover the pot with a lid and reduce the heat so the liquid simmers. Cook until the mussels open, 8 to 10 minutes. Without removing the lid, shake the pot once or twice during cooking to prevent burning. Once done, discard any mussels that have not opened.
Serve the mussels in the pot or transfer to a serving dish. Drizzle generously with the pan juices, and the sprinkle the bread crumbs on top.