Stir together the flour, sugar, salt, and yeast in a mixing bowl (or in the bowl of an electric mixer). Stir in (or mix in on low speed with the paddle attachment) the butter (or oil) and 3/4 cup (6 ounces, 170 grams) milk until the ingredients form a ball. If there is still loose flour in the bowl, dribble in some of the remaining 1/4 cup milk. The dough should be soft, tacky, and pliable, not stiff.
Sprinkle flour on the counter, transfer the dough to the counter, and begin kneading (or mixing on medium speed with the dough hook). Knead the dough for 8 to 10 minutes (or mix for about 8 minutes), sprinkling in more flour if needed to make a tacky, but not sticky, dough. It should pass the windowpane test and register 77˚F to 81˚F (25˚C to 27˚C). Lightly oil a large bowl and transfer the dough to the bowl, rolling it to coat it with oil. Cover the bowl with plastic wrap.
Ferment at room temperature for 1 to 1 1/2 hours, or until the dough doubles in size.
Wipe the counter with a damp cloth and transfer the dough to the counter. Divide the dough into 6 equal pieces of 3 ounces (85 grams) each. Shape the pieces into small balls. Line a sheet pan with baking parchment, mist the parchment lightly with spray oil, and dust with cornmeal. Transfer the dough balls to the sheet pan, spacing them about 3 inches (8 cm) apart. Mist them lightly with spray oil, sprinkle them loosely with cornmeal, and cover the pan loosely with plastic wrap or a towel.
Proof at room temperature for 1 to 1 1/2 hours, or until the pieces nearly double in size and swell both up and out.
Heat a skillet or flat griddle to medium (350˚F/177˚C if you have a thermometer setting). Also, preheat the oven to 350˚F (177˚C) with the oven rack on the middle shelf.
Brush the pan or griddle with vegetable oil or mist with spray oil. Uncover the muffin rounds, then slide a metal spatula under a round and gently transfer it to the pan. Add more rounds the same way, spacing the rounds at least 1 inch (2.5 cm) apart. Cover the rounds still on the sheet pan with plastic wrap or a towel to prevent them from developing a skin. The dough rounds will flatten in the hot pan and spread slightly and then they will puff somewhat. Cook the round for 5 to 8 minutes, or until the bottoms cannot cook any longer without burning. The bottoms should be a rich golden brown; they will brown quickly but will not burn for a while, so resist the temptation to turn them prematurely or they will fall when you flip them over. Using a metal spatula, carefully flip each one over and cook on the other side for 5 to 8 minutes in the same manner. Both sides will now be flat.
When the dough seems as if it cannot endure any further cooking without burning, transfer the pieces to a sheet pan and place the pan in the oven (don't wait for the still uncooked pieces, or the ones just out of the pan will cool down and will not respond to the oven stage). Bake for 5 to 8 minutes to ensure that the center is baked. Meanwhile, return to the uncooked pieces and cook them, then bake them, as you did the first batch.
Transfer the baked English Muffins to a cooling rack and let cool for at least 30 minutes before slicing or serving.