Combine water, yogurt, oil, yolk, and honey in a liquid measuring cup.

Process flour and yeast in a food processor to combine. With processor running, slowly stream in water mixture until just combined and no dry flour remains, about 10 seconds. Let dough rest 10 minutes.

Add 1 teaspoon salt to dough and process until a satiny, sticky dough ball forms and clears the sides of the bowl, about 60 seconds.

Transfer dough to a lightly floured surface and knead until smooth, 1 minute. Shape dough into a tight ball and place in a large, lightly oiled bowl. Cover tightly with plastic wrap and let rise at room temperature for 30 minutes.

After 30 minutes, fold partially risen dough over itself 8 times by gently lifting and folding edge of dough toward middle, rotating bowl 90-degrees after each fold. Cover with plastic wrap and let rise 30 minutes more. Repeat folding, then cover tightly and let rise one last time for 30 minutes.

Transfer dough to a lightly floured surface and divide into 4 equal pieces. Shape each piece into a smooth, tight ball and transfer to a lightly oiled baking sheet, spacing two inches apart. Cover loosely with lightly oiled plastic wrap and let rise 20 minutes.

Preheat oven to 200°F with a baking sheet set inside.

Transfer 1 dough ball to lightly floured surface and sprinkle with flour. Roll piece of dough into an 8–8½-inch round of even thickness, sprinkling with more flour as needed. Sprinkle ¾ teaspoon minced garlic over the surface of dough; gently press into dough. Using a fork, poke surface of dough 30 times.

Heat 1 teaspoon oil in a cast-iron skillet over medium until shimmering, wipe oil out of skillet with a paper towel.

Mist top of dough with water, then transfer to skillet, misted-side down; mist top surface of dough with water and cover. Cook dough until bottom is brown in spots across surface, about 2 minutes. Flip naan, cover and cook until golden brown in spots across surface, about 2 minutes.

Transfer naan to baking sheet in oven to keep warm.

Repeat rolling and cooking with remaining dough and oil.

Heat butter in a small saucepan over medium heat. Cook, swirling occasionally, until lightly golden and nutty, 3–5 minutes. Off heat, stir in remaining 1 teaspoon garlic and season with salt and pepper.