Grease a plate with unsalted butter. Set aside.
In 4-quart heavy nonstick saucepan, heat sweetened condensed milk, 1 tablespoon unsalted butter and coconut to boiling, stirring constantly with wooden spoon. Reduce heat to medium-low and continue to cook, stirring constantly, 10 to 15 minutes or until thick and shiny and starts to pull away from bottom and sides of pan. You will know mixture is ready when you tilt pan and mixture slides down easily. Transfer mixture to greased plate. Place in refrigerator or freezer until mixture is very cold.
Meanwhile, rinse strawberries and remove stems. Use cloth or paper towels to dry strawberries very well. They must be completely dry before you start making bonbons.
Grease hands with unsalted butter. Scoop up generous tablespoonful of cold coconut mixture; flatten in the palm of your hand. Place a strawberry over it and wrap coconut mixture around to conform to shape of strawberry. Repeat to use up coconut mixture. Place covered strawberries in freezer 15 minutes.
Meanwhile, use double boiler or microwave to melt dark chocolate. Cover cookie sheet with cooking parchment or waxed paper. Use dipping fork to drop covered strawberries one at a time into melted dark chocolate. Coat well with chocolate and then remove using dipping fork. Tap fork a few times against bowl to remove excess chocolate. Place back on cookie sheet, flat side down. Refrigerate until chocolate is set.
Use double boiler or microwave to melt white chocolate. Place melted white chocolate in freezer bag and cut off tip; drizzle chocolate over bonbons. Or place melted white chocolate in a squeeze bottle for drizzling. Refrigerate until chocolate is set.
To give this as a gift, place them in food gift boxes and decorate as you wish. Store bonbons in refrigerator and serve within 24 hours.