In a small bowl, stir the lemon juice, water, mirin, and soy sauce. Set aside.
Sprinkle the eggplant lightly with salt and let sit for 20 minutes. Rinse and pat dry. Dry all the other vegetables and set aside. Dry the shrimp well, season with salt and pepper, and set aside.
Pour 3 to 4 inches of oil into a deep, heavy pot. Begin heating the oil to between 350° and 360°F.
In a medium bowl set over an ice bath, combine the flour and cornstarch; whisk to blend. Whisk in the club soda gradually.
Continue whisking until the batter is smooth, and then add the herbs and sesame seeds. The batter should coat the back of a spoon as a custard would; your finger pulled across the spoon should leave a clear line. If it’s too thin, whisk in 1 to 2 Tbs. more flour and add more herbs and sesame seeds, if you like.
Set up a work area next to the oil: the vegetables and shrimp on the far side, the bowl of batter over an ice bath, and then the hot oil, which should be up to temperature. On the other side, set a tray or plate lined with paper towels.
Dunk a vegetable or shrimp in the batter to coat well and, without shaking off excess batter, quickly but carefully lay it in the hot oil (the batter will drip). The oil may spatter; you may want to use tongs. Repeat with three to five more pieces and monitor the temperature of the oil.
Fry the tempura until the batter turns a very pale gold and the food is tender but still a little firm, 2 to 3 minutes. If the pieces huddle together, separate them with a mesh skimmer. Turn the pieces if necessary.
With a mesh skimmer, transfer the tempura to a paper-towel-lined plate; sprinkle with salt and pepper. Continue dipping more pieces in the batter (stirring between rounds) and frying. Skim off batter particles from the oil between rounds.