Dissolve the yeast in the water; let stand for 3 to 5 minutes until a thin layer of tiny bubbles forms on top.
Put the sugar, both oils, salt, and flour in a stand-mixer bowl. Fit the mixer with a dough hook.
Add in the activated yeast mixture. Stir at low speed until the dough starts to form. Increase the speed to medium-high; knead the dough for 6 minutes until the dough is smooth. Remove the dough, coat the bowl lightly with olive oil. Return the dough to the bowl; cover with plastic wrap and let rise until almost doubled.
Punch the dough down; let rise again until almost doubled.
While the dough is rising the second time, shave the asparagus using a vegetable peeler. Grip the bottom of the asparagus and draw your peeler from your grip to the tip. Discard the tips and set the shaved asparagus aside.
Preheat the oven to 400º F.
Using a fine grater, grate the garlic into the oil. Stir to combine. (Alternatively, mince the garlic, stir into the oil, and blend with an immersion blender.)
Remove the dough to an oiled baking sheet. Press the dough into a 12-inch circle with your fingers.
Spread the shaved asparagus evenly over the top of your focaccia dough. Cover the dough with plastic wrap and let rise for 30 minutes. Remove the plastic wrap; spoon the garlic oil evenly over the top of the asparagus. Sprinkle evenly with coarse sea salt.
Bake the focaccia for 25 to 30 minutes. The top should be brown and the asparagus should be starting to char. Sprinkle with Parmesan and serve.
Notes
The rising time for the dough can vary greatly depending on the conditions in your kitchen. I usually find it takes between 45 to 1 hour for each rising.