Prepare the pickles: Bring vinegar, water, sugar, and salt to a boil in a small saucepan over high. Place carrots and radishes in separate bowls. Pour half of liquid into each bowl; chill at least 30 minutes or up to 12 hours.
Prepare the mussels: Heat oil in a large saucepan or Dutch oven with a tight-fitting lid over medium-high. Add shallots, lemongrass, ginger, and Fresno chiles. Cook, stirring often, until very aromatic and starting to toast, 4 to 5 minutes. Stir in coconut milk, vinegar, soy sauce, and sugar. Bring to a boil; stir in cilantro and mint. Cook 2 minutes. Add mussels, cover with lid, and cook until mussels open, 4 to 5 minutes. Discard any unopened mussels. Transfer opened mussels to a bowl. Pour cooking liquid through a fine wire-mesh strainer into a separate bowl; discard solids. Cool cooking liquid 10 minutes. Remove mussels from shells, and discard shells. Place mussels in cooking liquid; let stand at room temperature until ready to use (up to 30 minutes).
Stir together mayonnaise, lime juice, and shichimi togarashi in a bowl; set aside.
Heat a grill pan over high. Brush both sides of each bread slice with olive oil. Grill in grill pan until golden brown and toasted on both sides, 1 to 2 minutes per side. Remove mussels from liquid, and place on a plate. Spread 2 tablespoons mayonnaise mixture on each toast. Top with cucumber slices, mussels, pickled vegetables, cilantro, and jalapeño slices. Garnish each toast with a sprinkle of shichimi togarashi.