Barry Shaw had only acquired the female pair in the last month after moving into the house with his young family and creating the new home for the birds.
But the 21-year-old was shocked to discover the pets had been taken after he went to check on them at 11am yesterday.
Mr Shaw believes the birds, which he used while out hunting in Midlothian, were stolen for their value and would be sold on by the thieves.
Police were called to investigate the theft, and officers today appealed for anyone approached to buy the birds to come forward.
Mr Shaw, who lives in Park View in Loanhead, said: “I checked on them at 12.30am yesterday and they were fine but when I came back at 11am they were gone.
“I only moved into the house three or four months ago and I was happy because I knew I would have space to keep the birds in the back garden. There’s fields at the back of the house so it would be easy for someone to see them. I also have the hawks out on their perches during the day.
“I only go out hunting with a friend so no-one else would have known I had the hawks unless they saw them in the garden.
“I think they have been stolen for their value. They are each worth £350 and were not insured. I got one aged four from a man in Yorkshire and bought the three-year-old from a friend in Edinburgh. I’ve been checking the internet to see any sign of birds being offered for sale on sites like Gumtree.”
Mr Shaw, who is currently unemployed, said he used the birds for hunting rabbits, an activity he had enjoyed with his father since he was seven years old.
The aviary, which measures 8ft by 6ft, was in the garden of the home he shares with partner, Wendy Amos, 21, and their six-month-old son, Jaxson.
He added: “I’ve loved birds since I was a little boy. These birds had become much-loved members of the family in the time we’ve had them.
“I studied gamekeeping at college but it’s been difficult to get a job in that area. It’s something I still want to do, but going out with my hawks is a passion.”
The birds have dark brown feathers, with black tail feathers and white tips on their wings and tails. Both of them are officially registered with the Independent Bird Register and have blue rings.
A police spokeswoman said: “These birds are highly valuable, but would only be of interest to a very specialist market, so we would encourage anyone who is approached about them to contact us immediately.
“In addition, we would be keen for anyone who would have been in the area at the time, and who may have seen something suspicious, to get in touch on 0131-311 3131.”