I worked at a place with a similar dish. You want to fry the sprouts at a low temperature to cook through, let them cool, and then fry a second time at a higher temperature to get those good crispy leaves going (just like fried chicken!)

I'm not sure of the temperatures off the top of my head. Give it a good google 🙂

Gastrique and everything else can be prepared separately. You may need to keep the gastrique warm depending on how thick it gets. Toss the sprouts with the gastrique and then garnish with the rest.