Toss asparagus with olive oil and a generous pinch of Kosher salt. Place asparagus in a grill basket (or directly on the grill if the spears are large enough) and cook, turning occasionally, until asparagus is tender and slightly charred, about 5-8 minutes.
On the stove, heat olive oil in a cast iron skillet. Place asparagus spears into skillet, working in batches, and season generously with Kosher salt. Cook, turning occasionally, until asparagus is tender and slightly charred, about 8-10 minutes. Repeat with remaining batches.
In a small bowl, whisk together crème fraîche, horseradish, lemon juice, lemon zest, and freshly ground black pepper, and ½ the chives. Drizzle over asparagus spears and top with remaining chives.