Pour 3-1/2 cups boiling water over the bulgur in a medium bowl, stir, and let sit until the water is absorbed and the bulgur is tender, 20 to 25 minutes.
Transfer the bulgur to a foil-lined rimmed baking sheet, drizzle with 1 Tbs. of the olive oil, and toss lightly to coat. Spread bulgur on the baking sheet and cool completely at room temperature or in the refrigerator.
Put the lemon juice in a small bowl and gradually whisk in the remaining 1/2 cup of olive oil. Whisk in the tahini. Taste and season with salt, pepper, and additional lemon juice or olive oil as needed.
Put the cooked and cooled bulgur in a large serving bowl and toss to break up any clumps. Add the carrots, peppers, cucumbers, feta, parsley, dill and 1/2 cup vinaigrette and toss. Taste and season as needed with more vinaigrette, salt, and pepper, and serve.
The salad can be refrigerated for up to 1 day. If making ahead, let sit at room temperature so it’s not refrigerator-cold and season with more vinaigrette, salt, and pepper before serving.