Boil 5-6 eggs, leaving one for egg wash for bread. Cool completely. If desired, color eggs according to food-coloring package instructions. Eggs can be done ahead and chilled.
Heat milk in a small saucepan over medium heat till instant-read thermometer reads 110-115 degrees. Transfer milk to a 2-cup measuring cup; stir in 1 tablespoon sugar. Sprinkle yeast over milk and whisk to blend. Let sit until yeast is foamy, about 5 minutes. Add 2 eggs at room temperature; whisk until smooth.
Combine remaining 4 tablespoons sugar, flour, and salt in the bowl of a stand mixer fitted with a dough hook. Add milk mixture. With mixer running, add 1/2 cup room temperature butter, 1 piece at a time, blending well between additions. Mix on medium speed for 1 minute. Knead on medium-high speed until dough is soft and silky, about 5 minutes.
Brush a medium bowl with some melted butter; place dough in bowl. Brush top of dough with remaining melted butter; cover with plastic wrap. Let dough rise in a warm, draft-free area until doubled in size, 1-1 1/2 hours (or about 2 1/2 to 3 hours if made ahead and refrigerated).
Line a large baking sheet with parchment paper. Punch down dough. Divide into 2 equal pieces. With lightly floured hands, roll each piece on a lightly floured surface into a 16"--long rope with tapered ends (If dough 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 (braided loaf with be about 12" long). I angled it on the pan to give it more room. Tuck dyed eggs between braids, spacing evenly. Loosely cover with plastic wrap or a kitchen towel. Let rise in a warm, draft-free area until puffed but not doubled in size, 45-50 minutes.
Arrange rack in middle of oven; preheat to 375 degrees. Whisk remaining egg with 2 tsp warm water in a small bowl. Avoiding dyed eggs, brush dough all over with egg wash. Bake until bread is golden and an instant-read thermometer inserted into center of loaf registers 190 degrees, 20-25 minutes. Let cool on a wire rack. Serve warm or at room temperature, and be prepared to receive your compliments!