Melt butter in a pan over medium/low heat then add onions. Cook gently for 15 minutes, ensuring onions don’t become crispy.
Increase heat to medium and add half of the beer to the onions. Cook, stirring occasionally, until beer has evaporated. Add remaining beer and cook again until evaporated (about 15 minutes). Remove from heat.
Separate leaves from the Brussels sprouts by trimming the bottom stem and plucking off the leaves. When you can’t easily pull off anymore leaves, finely slice the small sprout core.
Toss Brussels sprout leaves and shreds with ½ Tbsp olive oil and a dash of salt and pepper. Use your finger to really work the oil into the leaf crevices for maximum crispiness while cooking!
Spread 1 Tbsp olive oil onto the dough, then sprinkle on onions, mozzarella, sprouts, and feta cheese.
Bake on a pizza pan or preheated pan at 425 degrees F (218 C) for 10 to 15 minutes, until top is lightly browned.
Drizzle with balsamic reduction and serve immediately.
*How to make a balsamic reduction: Bring ½ cup of balsamic vinegar to a simmer in a small saucepan. Whisking occasionally, allow balsamic to cook and reduce down until it is syrupy and coats the pan and spoon (it will reduce in volume by about half).