In a large bowl, combine the pancake mix, Mochiko flour, and salt.
Add the egg and about 1/3 cup of the milk. Mix until a soft dough forms, adding the remaining milk a little at a time as needed. The dough should be soft and pliable but firm enough to roll into balls without being overly sticky.
Transfer the dough to a work surface lightly dusted with Mochiko flour. Roll the dough into a long rope about 3/4-inch thick, then cut into 1-inch pieces. Roll each piece into a ball.
Cut parchment paper into individual squares. Arrange 6–8 dough balls in a ring on each piece of parchment, gently pressing the balls together. Dab the connecting points with a little water if needed to help them stick.
Heat about 2 inches of vegetable oil in a heavy-bottomed pot to approximately 325°F.
Carefully lower each donut into the oil with the parchment paper still attached. After a few seconds, once the parchment releases from the dough, carefully remove it with tongs.
Fry the donuts for about 2–3 minutes per side, or until puffed and evenly golden brown. Transfer to a wire rack and let cool slightly.
To make the salted caramel glaze, melt the butter in a small saucepan over medium-low heat. Add the brown sugar and heavy cream and stir until smooth. Bring to a gentle simmer and cook for 1–2 minutes, stirring constantly.
Remove from the heat and stir in the vanilla and a pinch of sea salt. Let the glaze cool for a few minutes until slightly thickened.
Dip the tops of the mochi donuts into the warm salted caramel glaze. Sprinkle with a little sea salt and let the glaze set before serving.