If not pre-cleaned, clean and prepare 1 lb squid: remove head, innards and cartilage; separate tentacles and slice tubes into 1/2-inch rings. Pat thoroughly dry with paper towels (very important for crispiness).

Marinate briefly

In a bowl whisk 3/4 cup buttermilk and 1 large beaten egg until combined. Add the dried squid to the buttermilk-egg mixture and toss to coat. Let rest 10–15 minutes in the refrigerator (this tenderizes and helps coating adhere).

Make the dry coating

In a shallow bowl or rimmed plate combine 1 cup all-purpose flour, 1/2 cup cornstarch, 1/4 cup semolina (if using), 1/2 tsp baking powder, 1 1/2 tsp salt, 1/2 tsp black pepper, 1/2 tsp garlic powder and 1/2 tsp paprika. Whisk to evenly distribute seasonings.

Dredge squid

Working in small batches, remove squid from the buttermilk mixture, letting excess drip back into the bowl. Place rings/tentacles into the dry mixture and toss to coat thoroughly, pressing the coating onto the squid so it adheres. Shake off excess. For extra crust, double-dip: after first dredge return briefly to the wet mix then dredge again in the dry mixture.

Heat oil

Pour about 3 to 4 cups vegetable oil into a heavy-bottomed pot or deep skillet so it reaches roughly 1 1/2 to 2 inches up the side. Heat over medium-high until oil reaches 350°F (175°C). If you don't have a thermometer, test with a small pinch of coating—if it sizzles and browns in ~30–45 seconds, it's ready.

Fry in batches

Carefully lower a single layer of coated calamari into the hot oil using a spider or slotted spoon. Fry in small batches for 1.5–2 minutes, turning once, until golden brown and just cooked through. Overcrowding lowers oil temperature and makes coating soggy. Remove with a slotted spoon and transfer to a wire rack set over a baking sheet or to paper towels to drain. Immediately season lightly with a pinch of salt. Repeat with remaining squid.

Keep warm and serve

If making multiple batches, keep cooked calamari in a 200°F (95°C) oven on the wire rack to remain crisp for up to 15 minutes. Sprinkle with chopped parsley and serve with lemon wedges alongside the aioli.

Aioli

Make aioli (by hand)

Place 1 room-temperature egg yolk in a medium bowl with 1 tsp (actually 1/2 tsp) Dijon mustard, 1 grated garlic clove, and 1 tbsp fresh lemon juice. Whisk together until combined. Very slowly begin to drizzle in the oils: first add the 3 tbsp extra-virgin olive oil drop by drop while whisking vigorously to emulsify; once it thickens, start adding the 3 tbsp neutral oil in a slow, steady thin stream while continuing to whisk until fully incorporated and thick. Season with 1/4 tsp salt and a pinch (1/8 tsp) black pepper, taste and adjust lemon or salt as needed. If the aioli becomes too thick, whisk in 1 tsp warm water to loosen.

Alternative: blender or food processor

In a small blender or using an immersion blender in a tall jar, combine the egg yolk, mustard, garlic and lemon juice. With the motor running, very slowly pour in the oils in a thin stream until the sauce thickens. Season to taste. Chill for 15 minutes before serving to let flavors meld.

Food safety note

This aioli uses raw egg yolk. If concerned about raw eggs, use pasteurized egg yolk or a prepared mayonnaise as a base (substitute 3/4 cup mayonnaise + 1 tsp lemon + 1 grated garlic clove).