In a large pot, bring 8 cups of water and 2 tablespoons salt to a boil. Remove and discard the bottom third of the broccolini stems. If some stems are very thick, cut them in half lengthwise.

Meanwhile, in a small bowl, whisk together the olive oil, balsamic vinegar, mustard, garlic, 1 1/2 teaspoons salt, and the pepper. When the water comes to a full boil, add the broccolini, return to a boil, and cook over high heat for 2 minutes, until the stalks are just tender. Drain well and place in a large bowl. Pour enough dressing over the broccolini to moisten and toss well. Splash with a generous squeeze of fresh lemon juice, sprinkle with salt, and serve warm or hot.