Törökméz is a cute breakfast-all-day restaurant perched on the foot of Buda's Rose Hill. It being in Buda means that most of the customers are local Hungarians, and also that you'll need to cross the Danube to get to it from Pest.
Törökméz's trendy breakfast dishes are inspired from around the world: there's everthing from tasty huevos rancheros (€7) to acai bowl (€7), avocado toast (€7), Cuban sandwich (€5), and English breakfast (€8). If you'd prefer something more indigenous to the region, go for the snappy frankfurters (€5), or the bundáskenyér (€3), which is similar to a savory French toast.
Once here, it's worth visiting the scenic tomb of Gül Baba, located almost around the corner from Törökméz. Gül Baba was a Bektashi monk who died in 1541 when the Ottomans occupied Budapest, and today his tomb is the northernmost point of Islamic pilgrimage in the world, and also a scenic and peaceful area with sweeping views of Budapest.