- Preheat oven to 350°F.
- Remove stems and all outer leaves from artichokes.
- Pull out cone of tender inner leaves, and use a small spoon to scrape away prickly choke from center.
- Trim away any dark green bits from bottoms.
- Put trimmed artichokes into acidulated water.
- Cut squash in half, and remove seeds.
- Use a sharp knife to remove peel, and set squash aside.
- Melt butter in a saucepan over medium-high heat.
- Whisk in flour, and cook until lemon-colored and bubbly, for about 1 minute.
- Pour in milk, increase heat to high and bring to a boil, stirring constantly until smooth and thickened, for about 5 minutes.
- Reduce heat to medium-low and cook, stirring often, for about 10 minutes.
- Season with nutmeg, salt and pepper.
- Spread half of sauce over bottom of a medium-sized baking dish.
- Cut artichoke bottoms and squash into ¼-inch-thick slices.
- Arrange, alternating, on top of sauce.
- Pour remaining sauce over top, and spread out evenly with back of a knife.
- Sprinkle cheese over top.
- Bake until vegetables are softened and top is golden brown, for about 45 minutes.
- Remove from oven, and serve.