Combine milk, water, honey, and yeast in the bowl of an electric stand mixer. Let sit for 10 minutes.
Pour flour, oil, pumpkin puree, and salt into the mixture. Mix using a dough hook attachment until dough is soft, smooth, and tacky, 2 to 5 minutes. Add more flour or water if needed.
Place dough on a lightly floured surface and knead by hand for 3 to 4 minutes. Form into a ball and place into a large, greased bowl. Cover with plastic wrap and let rise until doubled in size, 1 1/2 to 2 hours.
Preheat the oven to 495 degrees F (255 degrees C). Place a large, cast iron Dutch oven into the oven to preheat.
Divide dough into 2 balls and flour lightly. Place 1 ball back into the greased bowl, cover with plastic wrap, and save in the refrigerator for a later time.
Place remaining ball onto parchment paper that has been sprinkled with cornmeal. Cover with plastic wrap and let rest for 30 minutes. Dust with flour and score.
Remove the preheated Dutch oven carefully from the oven and uncover. Lift parchment paper with dough and place into the Dutch oven. Trim any parchment paper that extends beyond the top. Cover carefully with the hot lid.
Bake in the preheated oven for 20 minutes. Remove lid and continue baking until golden, about 40 minutes more.
Remove from the oven and lift out bread carefully using the parchment paper. Let cool on a wire rack for at least 30 minutes before slicing.
Make grilled cheese sandwiches once the loaf has cooled enough to slice. Heat a skillet over medium heat. Spread mayonnaise over one side of each bread slice. Place bread mayonnaise-side down into the skillet. Spread Dijon mustard on the other side. Top with Cheddar cheese and Gruyere cheese slices. Cook for 2 minutes; sandwich both halves together. Cook until golden, 3 to 4 minutes per side.