For the dough: In a small bowl, combine the yeast with the sugar and 3/4 cup warm water. Let sit for 10 minutes until it begins to foam.
In a large bowl, combine the all-purpose flour, whole-wheat flour and salt and make a well in the center. Add the yeast mixture and the olive oil. Using a fork, incorporate the wet ingredients with the dry ingredients until a dough forms. On a lightly floured surface, continue to work and then knead the dough until it is smooth and elastic, about 8 minutes.
Lightly coat a clean bowl with additional olive oil and add the dough, turning once to coat the dough in oil. Cover and set in a warm place until doubled in size, about 1 hour.
Heat a grill pan over medium-high heat and preheat the oven to 400 degrees F.
Punch down the dough and divide it into 2 pieces, using 1 piece for the pizza. Wrap the remaining dough in plastic wrap and refrigerate or freeze for another pizza.
For the pizza: Stretch the dough into a 12-inch circle that will fit on the grill pan. Add the dough to the hot grill pan and cook until stiff and bubbles start to form, 1 to 2 minutes. Flip and cook on the other side until grill marks form and the dough is completely cooked through, 1 to 2 more minutes.
Transfer the crust to a baking sheet and top the pizza with the onions, potatoes, tomatoes and goat cheese. Add the loaded pizza to the oven and cook until the cheese melts, 6 to 8 minutes. Serve immediately.