Preheat the oven to 375 degrees. Place cupcake liners in all 12 cups of a muffin tin.
Melt the butter and brown sugar in a skillet over medium heat.
Add the kumquats and simmer for 5-6 minutes, turning once or twice. Remove from heat.
In a large bowl cream together the butter and sugar. Add in
the vanilla and eggs and mix well.
In a medium bowl combine the flour, baking soda,
cinnamon, and salt.
Add the flour mixture to the butter mixture and mix
until just combined.
Stir in the carrots, raisins, and walnuts and mix
until just combined.
Spoon a teaspoon of kumquat mixture into the bottom of each muffin cup.
Fill with batter ¾ of the way full.
Bake for 30 minute or until a toothpick inserted into the center comes out clean.
Remove from oven and cool completely on a wire rack.