Fill a sink with cold water and submerge asparagus to dislodge any sand. Break off pale, woody ends (approximately the bottom third of the stalk), and discard. Working from base to top, use a sharp paring knife to halve or quarter asparagus into thin strips.
Bring a large pot of water to a rapid boil. Salt water generously and cook linguine to al dente. Reserve pasta cooking water for plating.
Melt 1 tablespoon butter in a large, deep-sided skillet set over medium-high heat. Add half the asparagus, sprinkle with sea salt, and cook just until spears go slightly limp and are a vibrant green, 3 to 4 minutes. Remove to a plate and cook the second batch.
Melt another tablespoon of butter in the pan, add half the shrimp, sprinkle with sea salt, and cook just until shrimp pinks up and the skin edges golden, about 2 minutes per side. Remove cooked shrimp to a plate and cook the second batch.
Turn heat to low, add remaining 2 tablespoons butter to the pan, and when melted, stir in garlic and chives; sauté for 1 minute. Stirring constantly, pour in half and half. Sprinkle in Pecorino, stirring until cheese has melted. Fold in cooked asparagus and shrimp.
To serve, twist linguine into shallow bowls, top with cream sauce, using reserved pasta water to loosen the sauce if needed. Top with asparagus and shrimp. Finish dish with chive blossom petals and a twist of pepper. Serve immediately.