Pulse carrots in a food processor until finely chopped, then transfer to a mixing bowl.
Blend green onions and tamari until minced, then add to bowl.
Combine tofu, coconut oil, nutritional yeast, lemon juice, mustard, olive oil, sesame oil, rice vinegar, onion powder, garlic powder, salt, and black pepper in food processor. Blend on high until very smooth, scraping down sides periodically.
Add tofu mixture and poppy seeds to bowl. Mix until thoroughly combined.
Spread evenly in serving bowl. Top with sesame seeds & poppy seeds.
Refrigerate at least two hours, or preferably overnight, before serving.
Notes
If using a small food processor, work in batches to make spread very smooth.