For garlic: In a small saucepan, place garlic. Add sufficient olive oil to submerge garlic. Carry to a boil over medium warmth; scale back warmth, and simmer, stirring incessantly, till garlic could be very tender, about half-hour. Let cool utterly.
Crush roasted garlic in oil; switch combination to a resealable jar, and refrigerate for as much as 2 weeks.
For onions: In a small skillet, soften butter over medium warmth. Add onion, sugar, and kosher salt; cook dinner, stirring incessantly, till onion is tender and translucent, about 10 minutes. Cut back warmth to medium-low; cook dinner, stirring incessantly, till golden brown, 25 to 35 minutes. Let cool utterly. Refrigerate in an hermetic container for as much as 3 days.
For dough: Within the bowl of a stand mixer, mix 1 cup plus 3 tablespoons (270 grams) room temperature water and 1½ tablespoons (21 grams) extra-virgin olive oil. Add flour, kosher salt, and yeast. Utilizing the paddle attachment, beat at low pace till mixed, about 1 minute; scrape sides of bowl and paddle. Beat at medium-low pace till dough pulls away from sides of bowl and varieties a clean ball on paddle, 3 to 4 minutes, stopping to scrape sides of bowl.
Oil a big bowl. Place dough in bowl, turning to grease prime. Cowl and let rise in a heat, draft-free place (75°F/24°C) for 1½ hours, folding dough in bowl each half-hour. (To fold, use moist or oiled fingers to succeed in below one facet of dough; gently stretch up, and fold over heart. Repeat 3 extra occasions round bowl.)
Spray a 12-cup muffin pan with cooking spray.
Prove dough onto a calmly floured floor, and divide into 12 parts (about 58 grams every). Place every portion right into a ready muffin cup. Cowl with plastic wrap, and let rise in a heat, draft-free place (75°F/24°C) till puffed and dough simply reaches prime of muffin cups, about 20 minutes.
Preheat oven to 425°F (220°C).
Drizzle remaining 1½ tablespoons (21 grams) extra-virgin olive oil onto rolls. Utilizing your fingers, dimple dough, touching pan via dough with out tearing dough.
Dot or unfold about 1 crushed roasted garlic clove on every roll. Season with sea salt (if utilizing).
Bake till tops are calmly browned and nearly set, about 5 minutes. High rolls with caramelized onion as desired. Bake till rolls are golden brown and an instant-read thermometer inserted in heart registers 190°F (88°C), 6 to 7 minutes extra. Let cool in pan on a wire rack for 10 minutes. Brush heat rolls with oil from roasted garlic. Garnish with rosemary, if desired. Serve heat. Retailer in an hermetic container for as much as 3 days.
