Stir together sugar, yeast, salt, baking soda, and 3 3/4 cups of the flour in the bowl of a stand mixer fitted with a dough-hook attachment.
Melt butter over low heat in a small saucepan. Remove from heat and add milk to pan. Add butter mixture and eggs to flour mixture; beat with stand mixer on medium speed until a shaggy dough forms, about 1 minute. Reduce mixer speed to low; beat until a smooth dough forms, about 5 minutes. If dough appears sticky, gradually beat in up to remaining 1/4 cup flour, 1 Tbs. at a time, until dough pulls away from bottom of bowl.
Place dough on a lightly floured work surface. Using your hands, knead dough into a smooth ball, about 1 minute. Return dough to bowl; cover and let rise at room temperature until doubled in size, about 1 hour 30 minutes.
Line a large baking sheet with parchment paper; set aside. Place dough on a lightly floured surface; roll into a 13-in. square.
Stir brown sugar with cinnamon in a small bowl. Spread softened butter evenly over dough; sprinkle evenly with brown-sugar mixture.
Roll dough into a log; place log seam side down. Cut log in half crosswise using a chef's knife. Slice each log half crosswise evenly into 6 rolls. Arrange 8 rolls on prepared baking sheet in the shape of a candy cane. Place remaining 4 rolls on the empty portion of the baking sheet.
Cover rolls with aluminum foil; refrigerate for at least 8 hours (overnight) or up to 2 days.
Preheat oven to 350°F. Remove rolls from refrigerator; let stand at room temperature while oven preheats.
Bake rolls, covered with foil, until puffed and firm but still pale, about 15 minutes. Remove foil; continue baking until lightly browned, 10 to 12 minutes.
Whisk cream cheese, vanilla, salt, and half (about 1/3 cup) of the confectioners' sugar in a bowl. Gradually whisk in remaining confectioners' sugar until just combined. Whisk in peppermint extract, if using. Gradually whisk in milk until smooth. Transfer half of the frosting to a small resealable plastic bag. Tint remaining frosting in bowl with red food coloring following package directions. Transfer tinted frosting to a separate small resealable plastic bag.
Move 8 rolls in cane shape to a platter and the extra 4 to a separate plate. Cut the tip off 1 corner of each resealable bag. Drizzle red and white frosting over rolls. Sprinkle with crushed-candy topping.