In the bowl of a stand mixer fitted with the paddle attachment, beat 1½ cups (188g) flour, sugar, yeast, and salt on medium-low speed until well combined.
In a medium saucepan, heat milk and ½ cup (113.5g) butter over medium heat until an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). Add milk mixture to flour mixture; beat at medium speed for 2 minutes. Add egg; beat at medium-high speed for 2 minutes. With mixer on low speed, gradually add 2¾ cups (344g) flour, beating just until combined and stopping to scrape sides of bowl.
Switch to the dough hook attachment. Beat at medium speed until a soft dough forms, 6 to 8 minutes, scraping down dough hook and sides of bowl as needed; add remaining ¼ cup (31g) flour, 1 tablespoon (8g) at a time, if necessary. (Dough should pass the windowpane test [see note 2], but may still stick slightly to sides of bowl.)
Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 40 minutes to 1 hour. (See note 3 for make-ahead method.)
Line a 13x9-inch rimmed baking sheet with parchment paper.
In a small bowl, stir together peppermint extract and remaining 1/2 cup (113.5g) butter.
Lightly punch down dough. Cover and let stand for 5 minutes. On a lightly floured surface, roll dough into an 18x12-inch rectangle. Using a small offset spatula, spread peppermint butter mixture onto dough, leaving a ½-inch border on one long side. Sprinkle with peppermint candies, white chocolate, and almonds. Starting with long side opposite border, roll up dough, jelly roll style; pinch seam to seal. Gently shape log to 18 inches long and even thickness, if necessary. Using a serrated knife dipped in flour, cut log into 12 slices (about 1½ inches thick), trimming edges slightly, if needed. Place slices, cut side down, in prepared pan. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 30 to 45 minutes.
Preheat oven to 350°F (180°C).
Bake until lightly golden and an instant-read thermometer inserted in center registers 190°F (88°C), 20 to 25 minutes, loosely covering with foil to prevent excess browning, if necessary. Let cool in pan for 30 minutes. Spread Cream Cheese Icing (see below) onto rolls; garnish with peppermint candies, if desired. Serve warm.
Make the cream cheese icing: In the bowl of a stand mixer fitted with the paddle attachment, beat butter, cream cheese, and salt at medium speed until smooth, 1 to 2 minutes, stopping to scrape sides of bowl. With mixer on low speed, gradually add confectioners’ sugar until combined; increase speed to medium, and beat until smooth and fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Use immediately.