A baraita[2] mentions cryptically that selecting (borer) some food from among other types of food is sometimes forbidden and sometimes permitted. The following three conditions, which different Amoraim[3] invoke to explain the baraita, are accepted by the Shulchan Aruch[4] as the halacha. Only if all of the conditions are present is it permitted to make the selection:[5] 1) The action is done by hand, not by a utensil whose purpose is selection. 2) The food that one wants to eat is removed from that which he does not now want. 3) The food that is removed will be used in the short term.
At first glance, your case fails the first test, as a utensil is being used, not hands. One could try to argue for leniency based on the following important rule that Rav Moshe Feinstein employed.[6] One is allowed to remove food that he wants to eat from its surroundings with a spoon or fork if the selection could have been done as efficiently by hand and the utensil was used for a side reason (e.g., to keep his hands clean). One could claim that in our case one would use his hands were it not for such factors as hygiene and not wanting to dirty or burn his hands. Then again, the spoon in question here is a special one that effectively is a strainer. It is likely, therefore, that in such a situation, Rav Feinstein would not have been lenient.
Nevertheless, we can permit using the slotted spoon in this context due to a combination of factors. First, the Maharitatz[7] says that the act of removing a solid from the medium of a liquid is not considered borer. He used this principle to explain his ruling that one may remove a fly that fell into a drink.[8] It is true that many disagree,[9] and, in fact, the more accepted halachic practice is to take out some liquid along with the fly.[10] On the other hand, the Yalkut Yosef[11] posits that the essential halacha follows the Maharitatz, just that it is preferable to remove some liquid with the fly. Furthermore, the Shevitat HaShabbat[12] maintains that when the solid pieces inside the liquid are large, even those who disagree with the Maharitatz would concede that removing large pieces is not borer.
Relating to your question, there is another significant reason to be lenient. The person who takes out the vegetables presumably does not care if a modest amount of liquid is transferred along with the vegetables. His intention is simply to efficiently move some vegetables from one place to another, not to remove liquid from the vegetables. Thus, even if liquid does fall off along the way, it is not considered borer. Based on this concept, Rav Ovadia Yosef[13] allows using a spoon with holes to remove pieces of meat from chulent, even though some gravy slips off in the process. He bases himself partially on a similar ruling in Shemirat Shabbat K’Hilchata.[14] There, Rav Neuwirth writes that one may use a ladle with holes in order to quickly transfer wet spaghetti from a pot to a plate when his intention is not for a significant amount of water to fall off in the process.
In summary, due to several reasons for leniency, it is permitted to use a slotted spoon to move vegetables (or other foods) from a pot of soup to bowls on Shabbat.