Preheat the oven to 200 degrees and turn on the grill setting.
Carefully rinse the asparagus tips and tomatoes under the tap. Leave the tomatoes on the sprigs. Pat dry with some paper towels.
Place a sheet of baking paper in a baking dish and place the asparagus tips and tomatoes in the dish. Drizzle with a little olive oil and season with salt and pepper.
Slide the baking dish into the oven and grill the vegetables for 15 minutes.
Meanwhile, make the dressing. Mix together the vegan mayonnaise, mustard, honey and lemon juice. Chop the dill and stir this in as well. Season the dressing with a little salt and pepper.
Tear the baby romaine lettuce leaves from the head and divide between a large bowl. Spoon some dressing on top.
Place the grilled vegetables on the lettuce leaves. Cut the red onion into fine half-moons and divide over the salad.
Top with a little more dressing and some Parmesan cheese. Serve the rest of the dressing separately for those who like extra dressing.