As MdBrnd said, there is no specific asthma specialty. However, asthma & COPD are one of the first diagnoses to be included in MTM guidelines since patient compliance, understanding & the ability to manage complex therapy does impact the course of their disease.
So...if pulmonary care is your interest, yes...you can involve yourself in that. However, you'll include COPD, chf, perhaps cystic fibrosis, lung cancer, etc....
You could do an inpatient primary care residency or an ambulatory care residency. My own opinion is you'll have a greater impact on folks with these diseases in an outpt setting rather than an inpt one.