For the beurre noisette: Melt butter over medium-high heat in a small saucepan. Foam will rise as the milk solids settle at the bottom. Reduce heat and skim off foam as needed. Keep butter on low heat until the solids begin to turn a light hazel color. Remove from heat and transfer to a separate bowl to prevent burning. Reserve.
For the pudding: Preheat oven to 350°F. Spread the coconut on a baking sheet. Toast in oven until the coconut is golden brown – approximately 7-10 minutes. Cool to room temperature. Reserve.
Bring a ¼ cup of water to a boil in a medium saucepan. Add rice, cover, reduce heat, and simmer until all the water is absorbed – approximately 15 minutes.
Remove from heat and stir in 11⁄2 cups coconut milk, sugar, and salt. Return to heat and cook over low heat until creamy and liquid is absorbed – approximately 15 minutes. Remove from heat and let stand for 5 minutes, stirring frequently.
Gently fold in remaining 1⁄2 cup coconut milk, buerre noisette, vanilla extract, lemon zest, cardamom, and toasted coconut.
Divide rice pudding evenly into 6 serving glasses and serve with fresh fruit for a garnish if you wish.