Combine milk and butter in a small saucepan and heat just until butter melts. Place warm water in a small bowl, sprinkle yeast and sugar on top, and let stand until foamy and fragrant, about 5 minutes. Whisk together flour, coarse salt, and poppy seeds in a large bowl. Gather mixture into a large mound and create a well in the center. Pour milk and yeast mixtures into well. Gradually stir together mixtures with a wooden spoon, starting in center and working outward, until a dough forms.
Transfer dough to a lightly floured work surface and, with lightly floured hands, knead dough, adding more flour if necessary if dough is too sticky, until smooth and shiny, about 10 minutes. Cover dough with a lightly floured kitchen towel and let rise in a warm place until dough is doubled in size, about 2 hours.
Preheat oven to 500 degrees with a pizza stone placed on rack in lowest position. Meanwhile, punch down dough with lightly floured hands, cover with towel, and let rise again until doubled in size, about 1 hour. Divide dough into 4 equal portions. Working with 1 portion and keeping remaining portions covered, roll out dough as thinly as possible without tearing, about 1/8 inch thick, with a floured rolling pin. Prick dough all over with a fork and transfer to a lightly floured pizza peel or baking sheet. Lightly brush with oil and sprinkle with poppy seeds and flaky salt. Slide dough onto pizza stone and bake until dough bubbles and blisters in places and edges become crisp and golden brown, about 5 minutes. Repeat process with 3 remaining dough portions; serve warm.