In a large mixing bowl, beat the softened cream cheese, sugar, and both extracts together until smooth and creamy. Divide this mixture into four smaller bowls, adding different food coloring to each; aim for vibrant pastel colors. Once colored, refrigerate the fillings to firm them up while you continue with the dough preparation.

In a separate bowl, whisk together the flour, sugar, and salt until well mixed. In a small bowl, combine the active dry yeast with the warmed milk (approximately 100-110°F) and let it sit for about 5 minutes until frothy. Then, add the eggs and vanilla extract to the yeast mixture, stirring to combine before gradually incorporating the flour mixture to form a soft dough.

Once your dough begins to form, gradually add the softened unsalted butter, mixing until the dough becomes sticky and slightly elastic. You may choose to use a stand mixer with a dough hook for ease. After mixing, cover the bowl with a damp cloth and let it rise in a warm place for 1 to 1.5 hours, or until it has doubled in size.

Once risen, gently punch down the dough to release the air. Roll it out on a floured surface into a rectangle measuring approximately 12x16 inches. Lightly score the dough to mark sections where you will spread the cream cheese fillings. Use a spatula to spread each colored filling evenly across the dough, leaving a small border. Then, roll the dough tightly into a log shape and slice it lengthwise to create a beautiful twist effect.

Carefully transfer the twisted dough into two greased loaf pans lined with parchment paper. Cover with a soft towel and let the dough rise again in a warm environment for about 1 hour, until puffy and nearly doubled in size. This second rise allows the Easter Cream Cheese Babka to achieve its airy texture.

Preheat your oven to 350°F (175°C), ensuring it's fully heated before baking. Place the loaf pans in the oven and bake for approximately 30-40 minutes. Keep an eye on the bread; it’s done when the tops are golden brown, and an instant-read thermometer inserted into the center reads between 185-190°F.

As soon as the Easter Cream Cheese Babka comes out of the oven, brush the tops generously with the prepared sugar syrup for a glossy finish. Allow the babka to cool in the pans for 5 to 10 minutes before gently transferring it to wire racks. Let it cool completely before slicing and serving, ensuring the delightful swirls are showcased.