Halloween Spiderweb Cookies

Make thin icing: In large bowl, stir together 1 batch icing and 2 tablespoons water. Fit 1 pastry bag with #3-size tip and fill with thin icing.
Color thicker icing: Divide second batch icing between 2 medium bowls. To icing in 1 bowl, gradually mix in black food coloring until icing is black. Fit second pastry bag with second #3-size tip and fill with black icing.
Fit third pastry bag with #1-size tip and fill with remaining (thicker) white icing.
Pipe border: Using pastry bag with #1 tip and thicker white icing, pipe around outer edge of 1 cookie. Repeat with remaining cookies and let set 2 minutes.
"Flood" to fill center: Using pastry bag with #3 tip and thinner white icing, squirt pool of icing into center of 1 cookie, then spread to edge with small offset spatula. Repeat with second cookie, giving first time to set.
Pipe circles: Returning to first cookie, using pastry bag with black icing, pipe concentric circles over white icing, beginning in center and ending almost at edge. Repeat with second cookie, giving first time to set.
Form spiderweb pattern: Returning to first cookie, position tip of toothpick in center and drag through icing out to edge. Wipe tip and repeat 7 more times to make 8 evenly spaced, radiating lines like spokes of wheel.
Next, position tip at edge, halfway between 2 lines, and drag inward to center to create another line. Wipe tip and repeat 7 more times to make 8 more lines between first 8.
Repeat process with second iced cookie. Repeat with remaining cookies, working in batches of 2 to allow icing to partially set, but not harden, between steps. Let cookies dry, uncovered, at room temperature 10 hours or overnight. (Once dry, cookies will keep, layered between sheets of wax paper or parchment, in airtight container at room temperature 1 week.)