01 - Combine warm water, yeast, and granulated sugar in a large bowl. Let stand for 5 minutes until mixture becomes foamy, indicating yeast activation.
02 - Add bread flour, salt, cinnamon, and nutmeg to yeast mixture. Stir until shaggy dough forms and flour is fully incorporated.
03 - Turn dough onto floured surface and knead for 8-10 minutes until smooth and elastic, adding minimal flour only if sticky.
04 - Place dough in greased bowl, cover with plastic wrap or towel, and let rise in warm area for 1 hour until doubled in size.
05 - Beat softened cream cheese, powdered sugar, and vanilla extract until smooth and creamy. Refrigerate until ready to use.
06 - Punch down risen dough and divide into 8 equal pieces. Roll each piece into 12-inch rope. Flatten rope slightly, spread thin line of cream cheese filling along center, pinch edges to seal, and shape into ring, pinching ends together to close.
07 - Arrange bagels on parchment-lined baking sheet. Cover and let rise for 20 minutes until slightly puffed.
08 - Preheat oven to 425°F. Bring large pot of water to boil.
09 - Working in batches, boil bagels for 1 minute per side. Remove with slotted spoon and return to baking sheet.
10 - Whisk egg with 1 tbsp milk. Brush generously over each bagel.
11 - Bake for 20-25 minutes until golden brown. Cool on wire rack completely before icing.
12 - Whisk powdered sugar with 2-3 tbsp milk until thick, pourable consistency forms. Drizzle over cooled bagels and immediately sprinkle with purple, green, and gold sugars.