01 - Combine warm water, yeast, and granulated sugar in a large bowl. Let stand for 5 minutes until mixture becomes foamy and activated.
02 - Add bread flour, salt, cinnamon, and nutmeg to yeast mixture. Stir until a shaggy, sticky dough forms and all flour is incorporated.
03 - Turn dough onto a lightly floured surface. Knead vigorously for 8–10 minutes until dough is smooth, elastic, and springs back when pressed.
04 - Place dough in a greased bowl, turning to coat all surfaces. Cover with plastic wrap or damp towel. Let rise in warm place for 1 hour until doubled in bulk.
05 - Beat softened cream cheese, powdered sugar, and vanilla extract in medium bowl until completely smooth and creamy. Refrigerate until needed.
06 - Punch down risen dough to release air. Divide into 8 equal pieces. Roll each piece into a 12-inch rope, working from center outward.
07 - Flatten each rope slightly with fingers. Spread thin line of cream cheese filling along center length. Pinch dough firmly to seal, enclosing filling completely.
08 - Bring ends of each rope together to form a ring. Pinch ends tightly to seal. Place shaped bagels on parchment-lined baking sheet, spaced 2 inches apart.
09 - Cover bagels loosely with towel. Let rise for 20 minutes until puffy. Meanwhile, preheat oven to 425°F and bring large pot of water to boil.
10 - Working in batches of 3–4, carefully lower bagels into boiling water. Boil for 1 minute per side (2 minutes total). Remove with slotted spoon, drain briefly, return to baking sheet.
11 - Whisk egg with 1 tbsp milk until blended. Brush each bagel generously with egg wash using pastry brush for glossy finish.
12 - Bake at 425°F for 20–25 minutes until deeply golden brown and crusty, rotating pan halfway through. Transfer to wire rack to cool completely.
13 - Whisk powdered sugar with 2–3 tbsp milk to form thick, drizzle-able icing. Drizzle over cooled bagels. Immediately sprinkle with purple, green, and gold sugars while icing is wet.