Blend half the butter with the chili paste and 1/4 teaspoon salt in a mini food processor until smooth.
Cut a piece of parchment or wax paper about 12 inches long. Mound the butter on the parchment, shaping it into a rough 4 by 1-inch rectangle. Set aside.
Blend the remaining butter with the cilantro and 1/4 teaspoon salt in the mini food processor until smooth. Mound the cilantro butter lengthwise alongside the chili butter. Fold the parchment over the butter. Holding the bottom parchment down with one hand, press a wooden spoon handle, sharpening steel, or pastry scraper up against the butter, pressing the top paper into the bottom of the butter mound so the paper becomes taut against the butter, shaping it into a tight log. Twist the ends of the paper like a party favor. Refrigerate until firm, about 1 hour.
Slice the butter into pats, unwrap and serve.
NotesCook's Note: You can also store this compound butter in the freezer.