Broccoli, Mushroom, and Gouda Quiche with Polenta Crust

Crust Ingredients
(adapted from http://www.bojongourmet.com/2013/05/asparagus-and-shiitake-mushroom-tart.html)

1 cup vegetable stock
1/4 cup heavy cream
1/4 teaspoon fine sea salt
1/2 cup polenta (or yellow corn grits)
1/2 tablespoon unsalted butter, plus more for greasing the pan
1/4 cup (0.75 oz) grated parmesan
1/8 teaspoon freshly ground black pepper
1/2 large egg

Filling Ingredients
(adapted from http://www.chow.com/recipes/11603-broccoli-mushroom-and-gouda-quiche)

2 cups large dice broccoli florets and tender stems (about 6 ounces)
2 Tbs ghee
2 cups large dice portobello mushroom (from about 1 [4-ounce] mushroom)
1/2 medium red onion, small dice
3 large eggs
1 cup half-and-half
3/8 teaspoon salt
1/2 teaspoon freshly ground black pepper
1 cup shredded young Gouda cheese (3.2 - 4 oz)

Prebake Crust
1. Preheat oven to 375 F.
2. Grease quiche dish with a thick layer of butter.
3. Bring stock, cream, and salt to a simmer.  Add polenta in steady stream, whisking constantly. Lower heat to medium-low and cook, stirring constantly, until polenta is very thick, about 10 minutes.
4. Off the heat, stir in butter, parmesan, and pepper, then stir in egg.
5. When cool enough to handle, use damp fingers to press mixture into dish.
6. Bake until surface is dry and pale golden, about 15-20 mins.

Make Filling
1. Bring a medium saucepan of heavily salted water to a boil over high heat. Once boiling, add broccoli and cook until fork tender, about 2 minutes. Drain and let cool in a single layer on a large plate.
2. Heat ghee in a medium frying pan over medium-high heat. When it shimmers, add mushroom and onion and season with 1/4 tsp salt and freshly ground black pepper. Cook until vegetables are soft and mushroom edges are golden, about 5 minutes. Remove from heat and let cool slightly, about 10 minutes.
3. Distribute broccoli and mushroom mixture evenly in pie shell.
4. Whisk together eggs, half-and-half, 1/8 tsp salt, and pepper in a large bowl until incorporated and smooth. Stir in cheese.
4. Pour custard into pie shell (don't overfill or crust gets soggy) and bake at 350 F until puffed and golden brown, about 45 minutes. Let cool at least 20 minutes before slicing.