Combine fennel and vermouth in a glass container (save vermouth bottle). Cover and chill 2 days. Strain through a fine-mesh sieve back into bottle.
Bring rhubarb, sugar, and 1/2 cup water to a gentle simmer in a small saucepan over medium heat and cook until mixture is pink and slightly syrupy, 14–16 minutes. Let cool, then transfer to a blender and purée until smooth (you should have about 1/2 cup rhubarb purée).
Combine 12 oz. fennel-infused vermouth, rhubarb purée, lemon juice, gin, and 2 cups ice in a large pitcher. Stir until cold. Pour cocktail into ice-filled old-fashioned glasses and garnish with shaved rhubarb.
Vermouth can be infused and rhubarb purée can be made 1 week ahead; cover and chill separately.