INSTRUCTIONS
1. Heat oven to 450°. Halve garlic crosswise with a knife and brush with olive oil; wrap with foil. Roast until soft, 1 hour. Let cool and squeeze roasted garlic cloves from their skins into a bowl; mash with a fork to a paste.
2. Melt 2 tbsp. of the butter and add to paste. Then add milk, parsley, basil, salt, and eggs; stir until smooth.
3. Put flour into a large bowl; form a well in center. Slowly pour in the garlic–milk mixture, stirring with a fork to form a smooth batter.
4. Bring a 5-qt. saucepan of salted water to a boil over high heat. Set a perforated spätzle-making disk over the pot. Working in batches, scrape batter through holes into water. Cook until dumplings rise to surface, about 1 minute. Using a slotted spoon, transfer dumplings to a baking sheet.
5. Heat remaining 4 tbsp. of butter in a 12" ovenproof skillet over high heat. Add dumplings; cook, stirring, until lightly browned, 6–8 minutes. Meanwhile, heat broiler; put rack 5" from heating element. Sprinkle dumplings with cheese; broil until melted, about 2 minutes.