Of course there is; it's done every time you look at a db-generated page on this site.
<?
$query = 'select * from dogs';
if (isset($HTTP_GET_VARS['breed']))
$query = " where breed='" . $HTTP_GET_VARS['breed'] . "'";
echo $query;
?>
Make a web page out of that code and call it with/without ?breed=dachsund as an argument.
Now some "advanced" suggestions.
You should populate your dropdown selector with a database query, rather than hard-coding the breed options in PHP. You've apparently designed your database as a single table, so can do that with SELECT UNIQUE. However ...
Your table design reflects a common mistake. Ideally you should have a table of dogs, and a table of breeds, with breed-<dog as a one-to-many relationship.
Here's why. What if you wanted to print out information about the breed -- how big does it get, how much food does it typically consume, what's the latin name, is it good with children, does it shed, et cetera? For that matter, what happens when people start misspelling "dachshund" when the data is entered into your tables? A separate breed table, linked to the dog table with a relational reference (foreign key) will prevent that.
Think about a query like this: Find me all the dogs that are good with children, have been here more than 14 days, are female, and grow to be 20-30 pounds in weight. Common breed characteristics belong in the common breed table; specific dog characteristics belong in the dog table.