Heat the oil in a medium saucepan over medium heat. Add the shallots and the garlic and cook until soft, about 1 minute. Increase the heat to high, add the wine, bring to a boil and cook until reduced by half. Add the cream and slowly start whisking in the cold butter, piece by piece and continue to whisk until the sauce is emulsified. Strain into a bowl and stir in the chives. Season with salt and pepper, to taste, and set aside.
For the shrimp:
Whisk together the oil, paprika, garlic and thyme in a large bowl; add the shrimp and toss to coat the shrimp in the mixture. Cover and let marinate in the refrigerator for 30 minutes.
Heat the grill to high.
Season the shrimp with salt and pepper and grill until slightly charred and just cooked through, about 2 1/2 minutes per side. Put some of the sauce on a large platter, drizzle with Basil Oil and top with the shrimp. Drizzle with more of the sauce; garnish with chives and serve.