Preheat oven to 450F and bring a large pot of salted water to a boil.

Slice 4 squid into 1/4-inch-thick rings.

Combine crabmeat, salt, pepper, basil, tarragon, and shallots in a bowl.

Fill 4 uncut squid pieces with crabmeat stuffing, two-thirds full.

Close squid openings with toothpicks using a sewing stitch.

Place a large nonstick ovenproof pan over high heat and add olive oil.

When oil begins to smoke, place stuffed squid in pan and decrease heat to medium.

Cook until bottom of squid turns golden brown (3 minutes), then flip and add butter.

Cook for 2 minutes, then place pan in oven and cook for 2 minutes longer.

Remove squid from pan and place on a plate. Cover and keep warm.

Drain oil from pan and return to stove top over medium heat.

Add shallots, garlic, lemon juice, and white wine, and simmer until liquid evaporates (6 minutes).

Add cream and Parmesan cheese.

Cook for 1 minute, then stir in chives and reserved sliced squid; cook for 30 seconds.

Remove pan from heat and cover to keep warm.

Place a large pot of water over high heat and bring to a boil.

Add pasta and cook for 2 minutes.

Drain pasta.

Return pan with sauce to medium heat, add pasta, and season with salt and pepper.

Cook until pasta is well combined (1 minute).

Divide pasta among 4 plates.

Top each plate with one piece of stuffed squid, then spoon sliced squid and sauce over top.

Serve hot.