Adjust an oven rack to the middle position and heat the oven to 300 degrees F.

Place the breadcrumbs in a medium bowl and toss with 2 tablespoons of olive oil and ¼ teaspoon salt. Transfer to a rimmed sheet pan and spread in an even layer. Bake until the breadcrumbs are lightly golden (10 minutes or so), stirring midway through. Transfer them to a large plate and set aside.

Increase the oven temperature to 375 degrees.

Line the sheet pan with aluminum foil and set the eggplant on top. Prick it 3 to 4 times with a fork and roast until a pairing knife easily slips into the center (40 to 50 minutes). It should be very tender throughout. Remove the eggplant from the oven and use scissors to cut an X in the bottom. Transfer it, stem side up, to a colander and set in the sink and let it drain and cool for 20 minutes.

Set the eggplant on a cutting board and slice it open lengthwise and scoop out the flesh and place it in the bowl of a food processor. Add the pureed carrot (or egg), lentils, garlic, basil, parsley, and nutritional yeast (or chickpea flour) and ¾ teaspoon of salt, black pepper, and crushed red pepper flakes. Process for 12 one-second pulses to combine. Add the bread crumbs and pulse 2 to 3 times to combine.

Scrape the mixture in a medium bowl, cover with plastic wrap flush against the surface, and refrigerate for 20 minutes or overnight.

Shape the eggplant mixture into golf ball-size pieces and roll them until they are nice and round.

Heat the grapeseed oil and remaining ½ cup of olive oil in a large heavy-bottomed skillet over medium-high heat for 2 minutes.

Drop one of the balls into the oil, it should immediately sizzle—if not, let the oil heat some more. Add a few more balls to the oil, taking care not to overcrowd the pan. Fry the 'meatballs' in batches, browning them on all sides (3 to 5 minutes for each batch).

Transfer to a paper towel-lined plate and serve immediately.