Rinse broccoli. Trim off leaves and stem ends. Cut crowns into 1- to 1 1/2-inch florets, and thinly slice stalks crosswise.
In a 5- to 6-quart pan over high heat, bring about 3 quarts water to a boil. Add pasta and cook, uncovered, stirring occasionally, until barely tender to bite, 7 to 10 minutes.
Meanwhile, in an 11- to 12-inch frying pan over medium-low heat, stir garlic, ginger, and cumin in peanut oil until garlic is soft, 3 to 5 minutes. Increase heat to medium. Add broccoli and 1/4 cup water. Cover and cook just until broccoli is tender-crisp to bite, about 2 minutes.
In a small bowl, mix coconut milk, tahini, soy sauce, and sesame oil. Add to broccoli mixture; stir often just until simmering.
Drain pasta and return to pan. Gently stir in broccoli mixture. Spoon into a serving bowl. Serve with hot sauce, lime wedges, and salt to add to taste.