Preheat oven to 400°F (200°C).

Bake until deep golden brown, about 40 minutes. Unwrap garlic, and let cool completely.

Squeeze roasted garlic cloves into a small bowl, and mash into a paste with a fork. (You should have about 2½ teaspoons or 25 grams.)

For dough: In the bowl of a stand mixer fitted with the paddle attachment, combine 1½ cups (188 grams) flour, sugar, kosher salt, and yeast.

In a small saucepan, heat garlic paste, mashed potatoes, milk, sour cream, and butter over medium heat, stirring occasionally, until an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). Add hot potato mixture to flour mixture, and beat at medium speed until combined. Beat in egg until combined. With mixer on low speed, gradually add remaining 3¼ cups (406 grams) flour, beating just until a shaggy dough forms.

Switch to the dough hook attachment. Beat at low speed until a soft, somewhat sticky dough forms, 15 to 20 minutes, stopping to scrape dough hook and sides of bowl. Cover and let dough rest for 15 minutes.

On a clean surface, divide dough into 36 portions (about 31 grams each). Using very lightly floured hands, roll 1 portion at a time into a smooth ball. (Keep remaining dough covered to prevent it from drying out.) Press 3 dough balls together, and place into each prepared muffin cup. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 40 to 45 minutes.

Preheat oven to 375°F (190°C).

In another small bowl, whisk together egg and 1 tablespoon (15 grams) water; brush onto dough. Sprinkle with sea salt.

Bake until golden brown and an instant-read thermometer inserted in center registers 190°F (88°C), 10 to 15 minutes, covering with foil to prevent excess browning, if necessary. Brush melted butter onto hot rolls, and garnish with chives, if desired. Let cool in pan for 10 minutes. Best served warm. Store in an airtight container for up to 3 days.