For the pickled vegetables: Combine the rice vinegar, sugar, kosher salt, star anise, and a few grinds of black pepper in a small saucepan. Cook over medium heat until the mixture is nearly boiling and the sugar is dissolved. Turn off the heat and stir in the daikon and carrot. Cool to room temperature, then refrigerate for at least 20 minutes or up to 1 day.
For the seasoned mayonnaise: Mix together the mayonnaise and hot sauce.
To complete the dish: Place a small skillet over medium-high heat. Add the vegetable oil and when it is hot, add the lemongrass, chanterelle mushrooms, and a pinch of kosher salt. Cook, stirring a couple of times, until the mushrooms are tender and lightly browned, about 3 minutes.
Toast the baguette slices until very lightly browned. They should still be more soft than crunchy.
Spread each piece of baguette with 1 teaspoon of the seasoned mayonnaise. Top with 1 slice of the pickled daikon and 1 slice of cucumber, followed by a few pickled carrots. Top that with 1 mushroom, a sprig of cilantro, a slice of jalapeño, and a few flakes of sea salt. Serve immediately.
Tip: Get Organize. There are a lot of ingredients to put on each crostini, so have them all lined up and ready to go. It will be more efficient if you first spread all the crostini with the mayo, then add the daikon to all of them and so forth.