There are two types of breast enlargement from hormone therapy (and no, they're not left, and right;-)
Men have the same breast buds that women had before puberty, but because they are exposed to more testosterone than estrogens in men, they don't grow into breast glands. The hormone therapy can mess with this and result in the breast buds seeing more estrogen than testosterone, and that can cause the glands to start growing. This is a condition called gynecomastia in men, and it is often accompanied by breast gland sensitivity/pain called mastodynia. Indeed, the mastodynia is often noticed first.
The other form of breast enlargement is breast fat development. Again, this is normally prevented by testosterone, but in the absence of testosterone, the body can redistribute fat in a female form which includes laying down more fat on the breasts. This is sometimes called pseudogynecomastia. There is no mastodynia in this case.
gynecomastia and mastodynia are more common in men who are only on Bicalutamide as their hormone therapy, and pseudogynecomastia is more common in men on hormone therapy injections (with or without Bicalutamide). However, they can both happen in both cases. Some men also report a brief spell of gynecomastia and mastodynia when their testosterone returns after finishing hormone therapy injections - the cause of this is unknown but is probably the same as can happen in some boys starting puberty.
A single (or occasionally double) radiotherapy blast to the breast buds aims to destroy the ability of the breast buds to grow. It is often said this would need to be given before any breast bud growth had started. Some research shows this treatment has around a 50% success rate. It does come with a little risk of causing secondary breast cancer in the future, and also of causing minor damage the heart muscle (many cases are incidental findings when looking for something else, but are not causing any significant side effects). When this treatment works, it prevents subsequent gynecomastia, but it has no impact on pseudogynecomastia. Not all centres do this treatment, and in practice it often can't happen because urology start you on hormone therapy before you get to see oncology, by which time it's probably too late to have the treatment.
Another treatment to prevent gynecomastia and mastodynia, much more popular, and around 70% effective is the drug Tamoxifen. This is a selective estrogen blocker, and stops breast tissue from being able to see estrogens, so it's more commonly used as part of the treatment of hormone sensitive breast cancer. For men on Bicalutamide alone for longer than 2 months, it's sometimes prescribed with the Bicalutamide at the rate of 1 x 20mg/week. Alternatively, it can be added later when gynecomastia or mastodynia is experienced although a higher dose is likely to be required at least initially because it takes around 3-4 weeks to build up the working level in your system. It's often said you have to start Tamoxifen before gynecomastia starts but this isn't true, although it may be preferable to avoid needing a higher dose later. Tamoxifen can reverse very recent breast gland growth (again probably needing a higher dose), but not anything which is long-standing, although it would still help to prevent any additional growth. Tamoxifen has no impact on pseudogynecomastia.
Tamoxifen can be liver-toxic in some people, although this is more of an issue for higher doses used as part of breast cancer treatment, but you should get a liver function test done after 3-4 months on Tamoxifen, and then perhaps 6 or 12-monthly after that. Tamoxifen won't be available if you have any history or risk for DVT or heart attack, and it does have a DVT risk of its own.
The only preventative treatment for pseudogynecomastia is to avoid putting on excess fat through careful dieting.