For the sponge, heat milk in a microwave-safe measuring cup for 45 seconds to 1 minute, until very warm (120-130°F). Whisk together warm milk, sugar, yeast, and flour in a medium bowl; cover loosely with plastic wrap and let stand 15 minutes.
For the dough, combine sugar, cream cheese, butter, salt, vanilla, egg, and sponge in a large mixer bowl; beat at medium speed for 1 minute until combined. Add 2 cups (280 g) flour and mix in at low speed until a soft dough forms, about 2 to 3 minutes. Switch out the flat beater for a dough hook, and knead dough for 4 minutes until smooth and elastic, adding remaining 1/4 cup (35 g) of flour 1 tablespoon at a time to prevent dough from sticking. The dough is ready when it feels tacky but doesn’t stick to your fingers.
Place dough in a large bowl coated with nonstick spray and cover with plastic. Let the dough rise in a warm place until doubled in size, about 50 to 60 minutes. To check if dough has doubled, lightly flour two fingers and press them down into center of dough. If indentations remain, dough has risen enough.
While the dough rises, prepare the filling. Combine cream cheese, sugar, salt, Meyer lemon juice, and vanilla in a food processor and process until smooth. Line a large rimmed baking sheet with parchment paper and dust lightly with flour. Gently punch down dough and roll out into a 12 x 15-inch rectangle on prepared baking sheet.
Use an offset spatula to spread cream cheese mixture down the center of the dough, leaving a 4-inch border on each side and a 1-inch border at the top and bottom. Spread Meyer lemon curd over the cream cheese filling. Use a sharp knife or a pizza cutter to cut 15 1-inch strips on both sides of dough (be careful not to cut parchment paper).
Remove the four corner strips and fold top border down over filling. Work from top to bottom and fold strips over filling, alternating strips diagonally (begin by folding over the top left strip, then fold the top right strip, then a left strip, and so on). When all the strips are folded over filling, fold up the bottom border and press to seal. Cover braid loosely with plastic and let rise 25 minutes. (Prepare ahead: After braid rests, cover and refrigerate overnight. Bring to room temperature before baking).
Preheat oven to 375°F. Bake braid until golden, about 25 minutes. Let cool on a wire rack for 15 minutes before glazing. For the glaze, whisk together powdered sugar, salt, melted butter, Meyer lemon zest, and juice in a small bowl. Drizzle glaze over braid and let set before serving.