In a large bowl, combine both lettuces with the shiso, tomatoes, dill, carrots, and cucumber. Using a knife, begin thinly slicing the chile from the tip. When you reach the seeds, stop slicing and discard or save the remaining chile for another use. Add the sliced chile to the lettuces in the bowl and toss to combine.
In a small bowl, whisk the olive oil with the lemon juice, rice vinegar, soy sauce, miso paste, sesame oil, and sugar. Using a Microplane set over a fine sieve set in a bowl, grate the ginger into the sieve, then press on the solids to drain as much juice as possible into the bowl. Pour 1 teaspoon of the ginger juice into the bowl with the dressing and discard the rest or save for another use. Season the dressing with salt and pepper and whisk until emulsified. Divide the dressing among 4 small ramekins and sprinkle each with some of the sesame seeds.
In a large pot of boiling water, cook the soba noodles until al dente, about 3 minutes. Drain the noodles into a colander and rinse under cold running water until the water runs clear. Drain the noodles again and divide among 4 serving plates. Divide the lettuces and vegetables over the noodles, sprinkle with the remaining sesame seeds, and serve immediately with the dressing on the side.