Bring 2 cups water to a simmer in a small saucepan and add the tea bags and 1 cup sugar. Simmer 3 minutes, then remove and discard the tea bags (to keep the syrup from getting too bitter). Boil the tea until reduced to a syrup, about 10 minutes. You should have about 1/3 to 1/2 cup tea syrup. Let cool completely.
Whisk together the flour, cinnamon, ginger, baking powder, allspice, cardamom, cloves, pepper and salt in a medium bowl. Beat the butter and remaining 1/2 cup sugar together in a bowl using an electric mixer on medium-high speed until light and fluffy, about 3 minutes. Beat in 1/4 cup tea syrup, then the egg and vanilla, until smooth. (Reserve the remaining tea syrup for another use; see Cook's Note.) Reduce the mixer speed to low and beat in the flour mixture until just combined. Divide the dough in half, wrap in plastic and form into two 7-by-2-inch logs. Refrigerate until firm, at least 4 hours or up to overnight.
Preheat the oven to 325 degrees F. Line 2 baking sheets with parchment paper.
Slice the logs into rounds about 1/3-inch thick, discarding the very end pieces. Arrange on the baking sheets about 1 inch apart.
Bake the cookies, switching and rotating the baking sheets halfway through baking, until crisp and light golden on the edges, 14 to 16 minutes. Let the cookies cool completely on racks.
Melt the chocolate in a small bowl in a microwave on 50 percent power, stirring every 30 seconds, until melted. Stir until smooth. Pour the chocolate into a sandwich-size resealable plastic bag and snip a tiny piece off one corner with scissors. Drizzle the chocolate in a zigzag pattern over the cookies. Let the chocolate harden before serving.
Any extra tea syrup can be used like honey, such as for sweetening hot tea.