Make the coconut pastry cream. In a saucepan, whisk 1 cup coconut milk, 1/2 cup whole milk, 1/3 cup sugar, and a pinch of salt. Warm over medium heat until steaming. In a bowl, whisk 3 egg yolks with 3 tablespoons cornstarch until smooth. Temper by slowly whisking in half the hot milk, then return everything to the pan. Cook, whisking, until thick and bubbling. Remove from heat, stir in 2 tablespoons butter, 1 teaspoon vanilla, and 1/2 cup toasted shredded coconut. Transfer to a bowl, press plastic wrap directly on the surface, and chill until cold.
Bloom the yeast. Warm 3/4 cup milk to about 105–110°F. Stir in 2 teaspoons sugar and 2 1/4 teaspoons active dry yeast. Let sit until foamy, about 5–10 minutes. (If using instant yeast, you can skip blooming and mix it with the flour.)
Make the dough. In a mixer bowl, combine 2 3/4 cups flour, remaining sugar (about 1/4 cup), and 1/2 teaspoon salt. Add the yeast mixture, 2 beaten eggs, and 4 tablespoons softened butter. Mix on low until a shaggy dough forms, then knead 6–8 minutes until smooth and slightly tacky. Add a tablespoon of flour at a time if overly sticky.
First rise. Shape into a ball, place in a lightly greased bowl, cover, and let rise in a warm spot until doubled, about 1–1.5 hours.
Shape the doughnuts. Roll the dough on a lightly floured surface to 1/2-inch thickness. Cut rounds with a 3-inch cutter. Gather scraps, rest 10 minutes, and re-roll to cut more. Place rounds on parchment-lined sheets, leaving space.
Second rise. Cover lightly and let rise until puffy, 30–45 minutes. They should look airy and jiggle slightly when nudged.
Heat the oil. In a heavy pot, heat 2–3 inches of oil to 350°F. Keep a thermometer clipped to maintain temperature.
Fry. Carefully lower 3–4 dough rounds at a time into the oil. Fry 1–2 minutes per side until deep golden with a pale ring around the center. Adjust heat to hold 350°F. Transfer to a rack to drain and cool.
Make the glaze. Whisk 1 1/2 cups powdered sugar with 2–3 tablespoons milk and 1/2 teaspoon vanilla until smooth and pourable. Add a splash more milk if too thick.
Fill the doughnuts. Poke a hole in the side of each doughnut with a paring knife or skewer. Spoon the chilled coconut cream into a piping bag fitted with a small round tip. Pipe filling into each doughnut until it feels slightly heavier and a bit of cream peeks out.
Glaze and finish. Dip the tops in glaze, let excess drip off, then press into shredded coconut. For Easter flair, add a few pastel sprinkles or perch a tiny chocolate egg on top.
Let set and serve. Allow the glaze to set for 10–15 minutes. Enjoy slightly warm or at room temperature.