1/2 cup unsalted butter, cut into 1-inch pieces, room temperature, plus 1/2 TBSP melted
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Steps
Heat milk in small saucepan over medium heat or in a microwave until an instant-read thermometer registers 110-115 degrees. Transfer milk to a2 cup measuring cut; stir in 1 TBSP sugar. Sprinkle yeast over milk and whisk to blend. Let sit until yeast is foamy, about 5 minutes. Add 2 eggs; whisk until smooth. COmbine remaining 4 TBSP sugar, flour and salt in the bowl of a stand mixer fitted with a dough hook.
Mix in milk mixture to dry ingredients. With mixer running, add 1/2 cup room-temperature butter. Mix on medium speed for 1 minute. Knead on medium-high speed until dough is soft and silky, about 5 minutes. Brush top of dough with remaining melted butter; cover with plastic wrap. Let rise to about double.
Line a large baking sheet with parchment paper (pr spray baking sheet). Punch down dough. Divide into three equal pieces. With lightly floured hands, roll each piece on a lightly floured surface into a 16-inch-long rope with tapered ends. (If fought begins to bounce back, cover with a kitchen towel and let rest for 10 minutes to allow dough to relax.)
Arrange ropes side by side lengthwise on prepared sheet. Pinch top ends together. Braid dough. Pinch bottom ends together to secure. The braided load will be about 12 inches long. Tuck dyed eggs between braids, spacing evenly. Loosely cover with plastic wrap or kitchen towel. Let rise in a warm, draft-free area until puffed but not doubled in size, about 45-50 minutes. Arrange a rack in the middle of oven; preheat to 375 degrees F.
Optional: whisk remaining egg with 2 tsp warm water in a small bowl. Avoiding dyed eggs, brush dough all over with egg wash.
Bake until golden brown. Let cool on a wire rack. Serve warm or at room temperature.
Tips
This can be made a day ahead. Cover dough with plastic; chill. Let dough rise in a warm, draft-free area until doubled in size, 1 to 1 1/2 hours (or 2-2/12 hours if dough has been refrigerated)