NULLs in a composite primary key - SQL Server
There is a lot of info missing in your question


Can a pet have multiple owners? eg we have a pet, not I have a pet
Do you expect more pet types (eg Rabbit, Bird, Fish)?


The problem with your current model (as shown) is the Pets table: it adds no value currently. Of course, your simplified design now hides information that probably makes it useful...

I'll throw in some ideas with assumptions though.

Note: This is correct design, whether EF can deal with it or not. That is, you design the model, implement the database, then make your client work with it

Option 1:

Assuming an owner can have 0..n dogs and/or 0..n pets, then you'd need 3 object tables and 2 link tables. You don't need Pet. 


Owner, Cat, Dog
Link tables for CatOwner and DogOwner


This allows multiple owners per Dog and Cat. You can of course just have an OwnerID in Dog and Cat to restrict to one owner.

You can have different attributes for Cat and Dog here.

As you add more types, this becomes unwieldy (no matter how you model owners)

Option 2:

Do you need to distinguish Cat or Dog as separate entities?
Isn't this just a type attribute of Pet?

**Pet
PetID, PK,
PetType, FK,
PetName
OwnerID, FK (optional, see below)
...


You can have a link table PetOwner (multiple owner) or OwnerID as an FK in Pet (single owner).

This requires each Cat or Dog (or Rabbit, Snake etc) object to have the same attributes.

Option 3:

The final way is superkey/subtype. 

This is is useful because if combines elements of options 1 (different attributes per Cat and Dog) and option 2 (keeps the idea of Pet)

This defines one Pet object that has exactly one sub type of Dog, Cat, Rabbit, Snake etc. The super key comes from the extra UQ1 in Pet to enforce subtype. Pet as a table also stores common attributes.

**Pet
PetID, PK, UQ1
PetType, UQ1, 'Dog' or 'Cat'
PetName
OwnerID, FK (optional, see below)
...

**Dog
PetID, PK, FK1
PetType, PK, FK1, constraint = 'Dog'
Breed  -- per Dog,not per Pet
CatsChased
...

**Cat
PetID, PK, FK1
PetType, PK, FK1, constraint = 'Cat'
Breed -- per Cat, not per Pet
MiceEaten
...


Note: UQ1 is a unique constraint used by the FKs in Cat and Dog. It is a "super key" on Pet that allows sub-types via the FK

Again, you can have a link table PetOwner (multiple owner) or OwnerID as an FK in Pet (single owner).

Note, PetType can be a constant derived column in Dog and Cat

Edit: added tables for Pet, Cat and Dog. 
I assume Owner is obvious...

CREATE TABLE PetSpecies (
    PetSpeciesID tinyint NOT NULL IDENTITY (1,1),
    Species varchar(100) NOT NULL,

    CONSTRAINT PK_PetSpecies PRIMARY KEY CLUSTERED (PetSpeciesID),
    CONSTRAINT UQ_PetSpecies UNIQUE (Species)
);
INSERT PetSpecies (Species) VALUES ('Dog');
INSERT PetSpecies (Species) VALUES ('Cat');

CREATE TABLE Pet (
    PetID int NOT NULL IDENTITY (1,1),
    PetSpeciesID tinyint NOT NULL,
    PetName varchar(100) NOT NULL
    OwnerID int NOT NULL,
    ...

    CONSTRAINT PK_Pet PRIMARY KEY CLUSTERED (PetID),
    CONSTRAINT FK_Pet_PetSpecies FOREIGN KEY (PetSpeciesID) 
                                 REFERENCES PetSpecies (PetSpeciesID),
    CONSTRAINT FK_Pet_Owner FOREIGN KEY (OwnerID) REFERENCE Owner (OwnerID),
    CONSTRAINT UQ_SuperKey UNIQUE (PetID, PetSpeciesID)
    );

CREATE TABLE Dog (
    PetID int NOT NULL,
    PetSpeciesID tinyint NOT NULL,
    CatsChased int NOT NULL,

    CONSTRAINT PK_Dog PRIMARY KEY CLUSTERED (PetID, PetSpeciesID),
    -- CONSTRAINT UQ_Dog UNIQUE (PetID),
    CONSTRAINT CK_Cat CHECK (PetSpeciesID = 1 /*dog*/),
    CONSTRAINT FK_Dog_Pet FOREIGN KEY (PetID, PetSpeciesID) 
                          REFERENCES Pet(PetID, PetSpeciesID)
    );

CREATE TABLE Cat (
    PetID int NOT NULL,
    PetSpeciesID tinyint NOT NULL,
    MiceEaten int NOT NULL,

    CONSTRAINT PK_Cat PRIMARY KEY CLUSTERED (PetID, PetSpeciesID),
    -- CONSTRAINT UQ_Cat UNIQUE (PetID),
    CONSTRAINT CK_Cat CHECK (PetSpeciesID = 2 /*cat*/),
    CONSTRAINT FK_Cat_Pet FOREIGN KEY (PetID, PetSpeciesID) 
                          REFERENCES Pet(PetID, PetSpeciesID)
    );


Notes: 


The PK of cat and dog can be PetID or PetID, PetSpeciesID. The latter is better to match the foreign key.
The unique constraint on cat and dog  is optional, really, as the index is contained within the PK and PetID is unique anyway if PetSpeciesID is constrained to a single value.