Whisk together all batter ingredients in a microwave safe bowl until smooth. The bowl should not be more than 2/3 of the way full of batter. The mixture will be similar in consistency to a cake batter.
Cover the bowl loosely with plastic wrap. (If the plastic wrap is too tight, condensation will not be able to escape and will drip back down into the mochi. It can also cause the batter to overflow!)
Microwave on high power for 1 minute, then stir with a heat proof non-stick utensil. When you stir it, it will be lumpy, and that’s okay! It doesn’t need to be smooth. Use oven mitts and be careful of steam when removing the plastic wrap and stirring.
Re-cover the bowl with the plastic wrap and continue microwaving and stirring one minute at a time until the mochi is no longer white, but has a more translucent quality and is very gummy. This usually takes about 2 more minutes, 3 minutes total.
Turn out the mochi onto a plastic cutting board or work surface that has been generously dusted with mochiko. Sprinkle more mochiko on top of the mochi. Pat the mochi out to a half inch thick disk. (Be careful, it will still be hot!)
While still warm but cooled enough to handle, shape your mochi. Use a knife to divide into four pieces. Fill each piece with 2 tsp of ube halaya or 1 tsp of halaya and a strawberry. Bring the corners of the mochi together over the filling, then use your fingers to pinch the remaining open seams closed. Roll the mochi in the coconut to coat.
Mochi can be kept at covered room temperature for several hours, but is best eaten fresh! Refrigeration is not recommended because the fat in the coconut milk will cause the mochi to lose its tender texture.