Monte Cristo Strata

2 tablespoons (1/4 stick) butter
12 ounces fresh mushrooms, sliced
18 to 20 slices quality white sandwich bread, crusts trimmed
10 ounces ham, cut into 1/2-inch pieces
10 large eggs
3 cups (packed) grated Gruyere cheese (12 ounce)
3 1/2 cups milk (do not use low-fat or non-fat)
2 tablespoons chopped fresh thyme or 2 teaspoons dried thyme
2 teaspoons dry mustard
3/4 teaspoon salt
1/2 teaspoon pepper

Melt butter in heavy medium skillet over medium-high heat.

Add mushrooms and saut