FOR PASTA: In a large bowl, whisk together flour and salt. Mound flour mixture and form a well in the center. Add milk and eggs to the well. Using a fork, gently break up yolks and slowly incorporate flour from inside rim of well. Continue until liquid is absorbed, then knead in bowl until dough forms a complete mass. Transfer to a well-floured work surface and knead for 3 to 4 minutes more. Wrap dough tightly in plastic wrap and let rest for 30 minutes. Meanwhile, return to filling.

 

Using shredding disc on a food processor or coarse holes of a box grater, coarsely grate beets. Heat butter in a large nonstick skillet over medium-high heat until melted; add beets, reduce heat to low and cook, stirring occasionally, until beets are dried out a bit, about 10 minutes. Transfer to a large bowl and let cool to room temperature. Add cheese, egg, salt and generous grinding of pepper; stir to combine. Stir in 1/4 cup breadcrumbs. If mixture seems damp, stir in remaining breadcrumbs.

 

Divide pasta dough into four pieces. Cover 3 pieces with plastic wrap. Flatten dough so that it will fit through the rollers of 

a hand-cranked pasta machine. Set rollers of pasta machine at the widest setting, then feed pasta through rollers 3 or 4 times, folding and turning pasta until it is smooth and the width of the machine. Roll pasta through machine, decreasing the setting one notch at a time (do not fold or turn pasta), until pasta sheet is about 1/16 inch thick.

 

Lay pasta sheet on a lightly floured work surface with the long side facing you; using a 2 1/2-inch round cutter, cut pasta into rounds. On 1 round, put about 3/4 teaspoon filling. Fold over round to form a half moon; moisten edges with a dab of water, press and crimp to seal and form decorative edge. Repeat with remaining pasta and filling. (The casunziei can be prepared and frozen for up to 3 months. To freeze freshly made casunziei, freeze them first on a parchment paper-lined baking sheet. Once frozen, transfer casunziei to well-sealed, airtight plastic freezer bags and keep frozen until ready to cook. Do not defrost before cooking.)

 

Bring a large wide pot of salted water to a low boil; avoid a rolling boil, which may cause the pasta to open up. In batches of 15 to 20, cook casunziei in water until they rise to surface, about 3 minutes (frozen pasta may take another minute or two longer). Using a slotted spoon, transfer casunziei to a large serving bowl. 

 

FOR SAUCE: In a small saucepan, heat butter over medium-high heat until melted and foamy. Pour butter over pasta, add cheese and toss to combine. Serve immediately, sprinkled with cheese and poppy seeds.