Place the butter in a food processor with the shallots, garlic, thyme leaves, anchovies, lemon zest and lemon juice. Puree until smooth.
With the motor running, drizzle the olive oil into the butter. Transfer the butter to a bowl and fold in the parsley and paprika and season with salt and pepper.
Roll up the butter in parchment paper to form a log and refrigerate until firm, about 2 hours, for easy slicing and portioning. The butter can also be frozen for up to 4 weeks.