For the breadcrumbs, place all the breadcrumb ingredients besides the olive oil in a bowl and mix to combine.
Heat the olive oil in a small skillet over medium heat. Once the oil is hot, add the breadcrumb mixture and cook, stirring frequently until the breadcrumbs are crispy and begin to turn golden-brown, 5 to 8 minutes. Remove from heat and set aside.
For the dressing, place its ingredients in a mason jar or bowl and shake or whisk until emulsified. Set aside.
(Note: When ready to serve, give it a good stir to recombine. If it’s too thick straight from the fridge, let it sit at room temperature for 10–15 minutes before using.)
To make the grilled cabbage, heat grill to medium-high heat.
Meanwhile, whisk together the olive oil, smoked paprika, garlic powder, salt, and pepper until combined.
Brush the cabbage wedges generously with the seasoned oil until thoroughly coated.
Grill, cut side down, until charred and slightly softened, 5 to 7 minutes per side.
Place the cabbage wedges on a serving plate. Garnish with minced parsley (if using). Drizzle on the aioli (or serve on the side for dipping), and finish with the Calabrian chili breadcrumbs. Enjoy!