To a pot of boiling salted water, add the soba noodles. Cook for 5 minutes and then give them a taste; you'll probably find they're just right. (Just FYI, I found that the box's instructions (cook for 8 to 10 minutes) yielded an overcooked noodle.) Drain and rinse the noodles in cold water and transfer the noodles to the refrigerator.
While the noodles are chilling, let's make the dressing. To a small bowl, add the tahini, sesame chile oil, honey and freshly zested ginger; whisk until smooth. Add the water and whisk more until very smooth. It might start to clump up at first, just keep whisking--it'll eventually become smooth. Also, you may need to add a bit of additional water until it becomes smooth (I only needed 2 tablespoons of water but you may need more). Adjust the seasoning by taste; adding a bit more chile oil, if you like, or more fresh ginger. Salt the dressing to taste (I found that I was happy with 1/4 teaspoon).
Remove the noodles from the refrigerator. If they're especially stuck together, you can add a bit of water to the noodles to loosen them up. Add the dressing, and using your hands really get in there to ensure that the noodles are thoroughly coated. Toss in the julienned carrots. Divide the noodles between plates.
To a small sauté pan, set over medium heat, add a tablespoon of olive oil. When the oil is hot, add the broccolini. Sprinkle with a few pinches of salt. Cook, tossing the broccolini, and cook for 2 minutes or so. Add the mushrooms and cook for an additional minute or so. Top each plate of soba noodles with a few broccolini and a few mushrooms. Garnish with sesame seeds, if you like.