Make a brine by combining the water, vinegar, fennel seeds, sugar, and kosher salt in a small saucepan. Place over medium high heat. Bring the brine to a simmer, stirring until the sugar is dissolved. Remove from the heat and let cool slightly.
Peel the carrots. To make ribbons, use a vegetable peeler to peel long strips from one side of the carrot. Pack the carrot ribbons into a jar. Add the chive blossoms and chive stems. Carefully pour the warm brine into the jar, pressing down to submerge the carrots as best you can.
Seal the jar and store in the fridge for at least 2 to 3 hours. The pickled carrots should keep for 1 to 2 weeks in the fridge. Drain the carrot ribbons before using.