My version of the class would be - 
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public class ProductBriefFutureHandler {

    @future
    public static void createProject(Set<Id> pbIds){
        List<Product_Brief__c> pbList = new List<Product_Brief__c>(); //this will hold the inserted or updated records
        Map<Id, Project__c> pbVsProjects = new Map<Id, Project__c>(); //this will help us map Projects to their Products
        //query the products
        pbList = [SELECT ID, Opportunity__r.ID, Opportunity__r.Owner.ID Select_BPC_Product_ASI__c, Related_Project__c FROM Product_Brief__c WHERE ID IN :pbIds];

        //iterate over each product and create a corresponding project
        for(Product_Brief__c pb : pbList){
            Project__c proj = new Project__c();
            proj.RecordTypeID = Project__c.sObjectType.getDescribe().getRecordTypeInfosByName().get('ASI').getRecordTypeId();
            proj.Related_Opportunity__c = pb.Opportunity__r.ID;
            proj.Sales_Rep__c = pb.Opportunity__r.Owner.ID;
            proj.Project_Description__c = pb.Select_BPC_Product_ASI__c;
            proj.Project_Category__c = 'ASI Bags and Tubing Sets';
            proj.Status__c = 'Active';
            proj.Start_Date_and_Time__c = System.Now();
            pbVsProjects.put(pb.Id, proj);    //put the product and associated project in the map
        }
        insert pbVsProjects.values();    //insert the projects, after insertion Id field would be populated in the inserted list
        
        //iterate over each product again to map to correct project
        for(Product_Brief__c pb : pbList) {
            pb.Related_Project__c = pbVsProjects.get(pb.Id).Id;    //get() shall give us the correct Project and then we need to traverse to Id
        }        
        update pbList;    //update the products
    }
}

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Please go through the comments to understand why its so. Forgive me for using Product instead of Product Brief.


I see the following challenges in this approach now - 

1. The update on Product Briefs in last statement of the future method would trigger the trigger again (since its on After Update event as well).
Solution would be the below change in trigger - 

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trigger ASIProjectCreation on Product_Brief__c (after insert, after update) {
    //skip the snippet below if context is Future i.e. trigger has been called from a transaction executing within a Future method
    if(!System.isFuture()){
        ProductBriefFutureHandler.createProject(Trigger.newMap.keySet());
    }
}
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We do not need the ProductBriefHandler class anymore, you may want to delete that.
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Everything should work fine now except following - 

1. You are creating a new Project everytime a Product Brief is being inserted or modified. This is a logical flaw.
There should be some criteria to determine when to create a new Project for example on update of Product Brief only when certain fields change in a specific way as your Product Brief already has an associated Project that you had created while creating the Product Brief itself.
Fix this logical flaw and you are rocking :)

2. In case you expect recursion for example say you create a Product Brief and then a workflow fires which does a field update on it, this should trigger your trigger again for update event (though it has already executed for insert and logically is not required to run again), you may want to introduce some Static variables to control the behavior (for time being you are good to go if you already don't have a recursive scenario in place).