Melt 2 tablespoons butter with 2 tablespoons olive oil in heavy large skillet over medium heat. Add onions, lower heat and sauté, stirring occasionally until tender and golden brown, about 45 minutes. Season with salt and pepper.
Melt remaining 2 tablespoons butter with 1 teaspoon olive oil in another heavy large skillet over medium heat. Add the reconstituted morels, garlic, and shallot.
Sauté briefly, stirring frequently until shallot is translucent. Add the wine and simmer until almost all of the liquid has been absorbed, stirring frequently, about 12 minutes. Season with salt and pepper.
Preheat oven to 500°F at least 30 minutes before baking. Preheat a large, heavy baking sheet or pizza stone in the oven.
Position rack in bottom third of oven.
Roll out dough to about 1/4 inch on a lightly floured surface, allowing dough to rest a few minutes if it springs back. Lightly brush dough with olive oil.
Scatter caramelized onions over the dough, then the morel mushrooms over onions. Scatter dollops of the goat cheese over the top.
Using a pizza peel or inverted baking sheet, slide the pizza onto the preheated baking sheet or stone (this gives it a crisper bottom crust).
Bake pizza 6 minutes, then rotate. Bake about 6 - 8 minutes longer, or until crust is golden brown.
Carefully transfer pizza to cutting board. Let rest 1 minute. Slice into wedges and serve.
Basic Pizza Dough
Makes 1 large or 6 individual pizza crusts.
1 1/2 cups warm water (110°F to 115°F)
1 1/2 Tbsp sugar
3 tsp active dry yeast
4 1/2 cups all purpose flour
4 1/2 Tbsp olive oil
2 1/4 tsp salt
Nonstick vegetable oil spray
In a large, heavy bowl,mix warm water and sugar. Sprinkle yeast in and stir to dissolve. Let stand until mixture bubbles, about 10 minutes.
Add flour, 4 1/2 tablespoons oil, and salt. Stir to mix thoroughly, about 1 minute.
Transfer dough to floured surface; knead until smooth, about 5 minutes.
Spray large bowl with oil. Transfer the dough to the prepared bowl, turning to coat with oil. Cover bowl with plastic wrap.
Let dough rise in warm draft-free area until doubled in volume, about 1 hour.
Punch down, cover with plastic wrap and refrigerate 1 hour before rolling and baking.