Preheat oven to 425 F (220 C). Line 1 or 2 baking sheets with parchment paper.
In a liquid measuring cup, stir vinegar/lemon juice into milk; set aside.
In a large bowl or bowl of stand mixer fitted with the paddle attachment, combine flour, sugar, baking powder, baking soda, and salt.
Cut butter into flour mixture using a pastry blender or paddle attachment on low speed until it forms pea-sized pieces.
Add shredded cheese and dill; mix until almost incorporated.
Pour in milk; mix until a soft, sticky dough forms.
Turn out dough onto a lightly floured surface and with floured hands knead until dough comes together (about 20 times), adding more flour to the dough if necessary.
Pat out dough into a 9-inch (23 cm) circle (3/4″ (1.5 cm) thick), cut into 12 wedges. Place on prepared baking sheet, at least 1-inch (2.5 cm) apart. Bake 15-20 minutes, or until puffed and golden.
Let cool for 10 minutes before serving. Serve with butter.