Cut tops off of 15 kumquats and remove any seeds with the tip of a knife. In a small saucepan, bring honey and 1 1/2 cups water to a boil. Reduce to a bare simmer.
Add kumquats and simmer for 10 minutes, stirring occasionally.
Pour kumquats in honey syrup mixture into a resealable jar and refrigerate overnight, or up to 2 weeks.
For the kumquat puree: Halve remaining 10 kumquats lengthwise and remove seeds with the tip of a knife. In a small saucepan, bring halved kumquats, remaining 1/2 cup water, and orange juice to a simmer.
Let simmer 10 minutes, remove from heat and let cool.
Place entire mixture in a blender and puree. Strain through a fine-mesh strainer into a resealable container. Kumquat purée keeps up to 5 days in the refrigerator and makes enough for several cocktails.
For each cocktail cocktail: In a mixing glass (or sturdy bowl), muddle the tarragon lightly. Fill shaker with ice.
Add clementine juice, blood orange juice, grapefruit juice, lemon juice, yuzu juice, kumquat puree, and white pepper to shaker and shake until well chilled, about 15 seconds.
Place one candied kumquat at bottom of glass. Fill glass with pebble ice and fine-strain juice blend into glass.
