Yes, although it was more common to put tiny songbirds in a pie rather than doves. A nursery rhyme reference to it:

Sing a song of sixpence,

A pocket full of rye.

Four and twenty blackbirds,

Baked in a pie.

When the pie was opened,

The birds began to sing;

Wasn't that a dainty dish,

To set before the king?

Chef Robert May wrote Accomplisht Cook in 1660 and you can read how it's done here.

The book Epulario, or The Italian banquet wherein is shewed the maner how to dresse and prepare all kind of flesh, foules or fishes. As also how to make ... profitable and necessary things (translated into English in 1598) contained such a recipe: ‘to make pies so that birds may be alive in them and flie out when it is cut up’ ....”

Make the coffin of a great pie or pastry, in the bottome thereof make a hole as big as your fist, or bigger if you will, let the sides of the coffin bee somewhat higher then ordinary pies, which done put it full of flower and bake it, and being baked, open the hole in the bottome, and take out the flower. Then having a pie of the bigness of the hole in the bottome of the coffin aforesaid, you shal put it into the coffin, withall put into the said coffin round about the aforesaid pie as many small live birds as the empty coffin will hold, besides the pie aforesaid. And this is to be done at such time as you send the pie to the table, and set before the guests: where uncovering or cutting up the lid of the great pie, all the birds will flie out, which is to delight and pleasure shew to the company. And because they shall not bee altogether mocked, you shall cut open the small pie, and in this sort you may make many others, the like you may do with a tart.