What kind of plant/mushroom is this?

Question

Two days ago I noticed this thing growing semi-covered in my garden near a grapevine:

As you can see, it was light purpalish - light pinkish kind of colour. On touch it was a bit soft, and the diameter of the whole thing was not more than 5-8cm. As far as I know, it looked like a mushroom.

Yesterday I decided to pick up this "mushroom" and slice it in order to try identifying it. I should say that removing this from the ground needed just a gentle pull.

Here is the unexpected result:

The slice showed the following (outer to inner):

1. Skin layer
2. Green jelly-like substance that was sticky to touch
3. White and hard to touch porous main part
4. Jelly-like white center

This thing does not look like anything I have seen before. I am no more even sure, that it was a mushroom.

Can you identify this?

I live in hardinest zone 6, Bulgaria. The thing was growing under a grape vine, so most of the day it was in shadow. It has been quite hot these days, around 30 C and above.

EDIT: To give you more sense of the size, one of these things is as wide as the circle you get by touching your thumb and index finger.

EDIT: After 36 hours in the composter, the white hard part has grown quite a bit. The plastic card on the photo is the size of a credit card.

EDIT: After consulting a Mushrooms Encyclopaedia, it turned out that this is actually Phallus hadriani. This corresponds to the observation Bamboo made for the root like thing - it is actually "aggregations of mycelium that resemble plant roots". Beside this, it appears that this species is endangered in Bulgaria... Next time I see it I'll just let it fruit and hopefully spread.

Answer 1

Maybe Phallus impudicus, or some other kind of Phallus. I'm from Germany and Phallus impudicus can be found here. It smells very bad. You could try to sniff them.

Answer 2

I don't think this is a fungi - in the third photo, showing it after you'd extracted it from the ground, there is a root showing which looks to be newly formed. This suggests to me that this might be some kind of seed case from something which has germinated and started growing. I do not, though, recognise what it is. If the root is intact, coming to a point at the end, then that might be what this is - if the root is broken, with more in the ground, then I'm not so sure. The other possibility is some kind of Crown Gall, but its not one I recognise.