Unroll the sheets of phyllo dough on a flat surface and cover them with a slightly damp towel. (If the towel is too wet, the dough will get sticky.) Working quickly so the dough doesn't dry out, place one sheet of phyllo on a board, brush lightly with melted butter, and sprinkle lightly with bread crumbs. Place a second sheet of phyllo on top, brush with melted butter, and sprinkle with bread crumbs. Continue until you have 4 sheets stacked up. Cut the sheets in half crosswise to make 2 almost-squares.
Cut one of the goat cheeses in half horizontally, making 2 thin discs. Place each disc in the center of one phyllo square. Starting at one corner, fold the phyllo up over the cheese and continue folding and pleating the phyllo around the cheese, as though you're wrapping a round gift that's tied on top. You will end with extra phyllo dough on top; twist the excess into a topknot. Brush the discs all over with melted butter and place on a sheet pan lined with parchment paper. Continue with the remaining phyllo dough and 2 goat cheese discs to make 4 additional packages.
Bake for 20 minutes, until the phyllo is lightly browned. Allow to cool for 5 minutes.
While the cheese packages bake, place the salad greens in a large bowl. Whisk the vinegar, mustard, garlic, salt, and pepper together in a small bowl. Still whisking, slowly add the olive oil, making an emulsion. Toss the salad greens with enough vinaigrette to moisten.
Distribute the salad among 6 plates and place one warm goat cheese package in the center of each plate. Sprinkle with salt and pepper and serve.