Melt butter over medium heat in a small saucepan until it browns and begins to smell nutty, about 3 minutes. Let cool.
In a blender, process dates, 1/4 cup almonds, and the vinegar to a coarse paste. Drizzle in browned butter, oil, 1/2 cup water, the salt, and pepper. Blend in 1 to 2 tbsp. more water if needed to thin.
Chop remaining 1/4 cup almonds. Put arugula, turnip, pear, and most of almonds and manchego in a serving bowl and toss with about 1/2 cup vinaigrette. Taste and add more vinaigrette if you like. Top with remaining cheese and almonds.
Make ahead: Vinaigrette, up to 2 days, chilled.