In a large bowl, thin the tahini with very hot water, adding by the tablespoon and whisking after each addition. (Some tahini is thick and will need 4 tbsp. of water.) Whisk in the tamari, ginger, garlic and chili paste.
In a large pot, bring a few inches of water to a simmer for the broccolini. Salt the water, add the broccolini and cook until crisp-tender, 3 to 4 minutes. Using a slotted spoon, transfer the broccolini to a bowl of ice water. Let cool briefly; drain. Fill the pot with fresh water and bring to a boil for the soba. Add the broccolini, scallions, sprouts, edamame and carrot to the tahini mixture.
Add the soba to the boiling water; cook according to package directions. Drain; toss with the vegetables and sauce. Sprinkle with sesame seeds.