In a microwave-safe bowl, heat the milk and butter until the butter is melted and the mixture is warm, about 60–90 seconds.

Add the salt and stir, then let cool until just warm to the touch (about 120°F (50°C)–130°F (55°C) / 49°C–54°C).

In another bowl, mix the warm water, about 1 tablespoon of the granulated sugar, and the yeast; stir and let stand 5–10 minutes until foamy.

To the yeast mixture, add the milk mixture, the remaining granulated sugar, eggs, and vanilla.

Stir in 1 cup of the flour until combined.

Keep adding flour about 1 cup at a time, mixing and kneading until a very soft dough forms and begins pulling away from the sides of the bowl; add the remaining ½ cup flour only if the dough is excessively sticky.

Knead with a dough hook on medium speed for about 5 minutes, or by hand for 8–10 minutes, until smooth and elastic.

Form the dough into a ball, place in a lightly oiled bowl, turn to coat, and cover.

Let rise in a warm place until doubled in size, 1–2 hours.

In a bowl, combine the brown sugar, granulated sugar, flour, cinnamon, and melted butter.

Mix until evenly blended and crumbly.

Set aside.

Line two baking sheets with parchment paper.

Punch down the risen dough and divide it in half.

On a lightly floured surface, roll one half into a rectangle approximately 10 × 16 inches.

Cut the rectangle lengthwise into two long strips.

Sprinkle the cinnamon filling evenly over both strips.

Roll each strip tightly into a log, seam side down.

Twist the two logs together to form a braid, then shape into an oval and pinch the ends to seal.

Transfer to a prepared baking sheet.

Repeat with the remaining dough and filling.

Cover loosely and let rise until puffy, 30–45 minutes.

Preheat the oven to 350°F (175°C).

Bake the king cakes for 25–30 minutes, until golden brown and baked through.

An internal temperature of 190°F (90°C)–195°F (88°C–91°C) indicates doneness.

Cool completely before frosting.

Whisk together the powdered sugar, milk, melted butter, lemon juice, and vanilla until smooth.

Adjust with additional milk if needed to achieve a thick but pourable consistency.

Insert a small plastic baby into the underside of each cake, if using.

Drizzle the frosting over the cooled cakes and spread gently if needed.

Decorate with purple, green, and gold sanding sugar while the frosting is still wet.

Slice and serve.