Preheat oven to 350 degrees F (175 degrees C). Place the halved acorn squash cut-side down in a glass baking dish; add 1/2-inch water.

Bake in the preheated oven until soft and tender, about 30 minutes. Remove squash from oven; scoop out flesh.

Place potatoes in a saucepan with enough water to cover; bring to a boil. Reduce heat and simmer, covered, until soft and tender, about 20 minutes. Drain potatoes.

Mash potatoes partially in a large bowl; stir in squash flesh, butter, mustard, dill, salt, and pepper. Add milk gradually, mashing to desired consistency.