1. Cook garlic, 1 tablespoon butter, and 1 teaspoon water in 12-inch cast-iron skillet over medium heat, stirring occasionally, until garlic is straw colored, about 2 minutes.
2. Add remaining 5 tablespoons butter, swirling to incorporate. Remove from heat and let mixture sit for 10 minutes.
3. Strain garlic butter through fine-mesh strainer into small bowl, reserving solids; do not clean skillet. 2 Whisk remaining ¾ cup water, yeast, 1 tablespoon garlic butter, and reserved garlic solids together in liquid measuring cup until yeast dissolves.
4. Whisk flour and salt together in bowl of stand mixer. Using dough hook with mixer on low speed, slowly add water mixture and mix until dough comes together, about 2 minutes.
5. Increase speed to medium and continue to mix until dough is smooth and elastic, about 8 minutes. (If after 4 minutes dough is still very sticky, add 1 to 2 tablespoons extra flour; dough should clear sides of bowl but stick to bottom.)
6. Transfer dough to lightly floured counter and knead by hand to form smooth, round ball, about 1 minute. Place dough in large, lightly greased bowl cover tightly with greased plastic wrap, and let rise until doubled in size, about 1 hour.
7. Transfer dough to lightly floured counter and shape into 12-inch log. Divide log into 12 equal pieces and cover with greased plastic. Working with 1 piece of dough at a time (keep other pieces covered), round dough into smooth, taut balls and arrange in now-empty skillet.
8. Cover rolls loosely with greased plastic and let rise until nearly doubled in size, about 1 hour.
9. Adjust oven rack to middle position and heat oven to 500 degrees. Brush tops of rolls with half of garlic butter, transfer skillet to oven, and bake until rolls are golden brown, about 12 minutes, rotating skillet halfway through baking.
10. Using potholders, transfer skillet to wire rack. Being careful of hot skillet handle, brush rolls with remaining garlic butter and let cool for at least 5 minutes before serving.