Preheat a grill to medium-high heat or heat a grill pan over medium-high heat. Grill the peppers, turning occasionally, until the skin is charred and blistered all over, about 20 minutes. Transfer to a medium metal or glass bowl, cover with plastic wrap and let the peppers steam for 10 minutes.
Meanwhile, place the asparagus, mushrooms and 3 tablespoons of the oil in a large mixing bowl. Season with salt and pepper and toss to combine. Grill the vegetables, flipping once, until they have lightly charred grill marks, about 1 to 2 minutes per side. Arrange on a serving platter with the broccoli and carrots.
When the peppers are cool enough to handle, peel off the skin, remove the seeds and coarsely chop the flesh. Transfer to a food processor, add the feta, yogurt, lemon juice, paprika and garlic and pulse until a thick paste starts to form. With the motor running, stream in the remaining 2 tablespoons oil until you have a smooth, thick dip. Adjust the seasoning, as needed. Transfer the dip to a serving bowl and serve with the crudite.