Use Future Methods

public class AccountProcessor {
    
    @future
    public static void countContacts(List<Id> accountId_lst) {
        
        Map<Id,Integer> account_cno = new Map<Id,Integer>(); 
        List<account> account_lst_all = new List<account>([select id, (select id from contacts) from account]);
        for(account a:account_lst_all) {
            account_cno.put(a.id,a.contacts.size()); //populate the map
            
        }
            		
        List<account> account_lst = new List<account>(); // list of account that we will upsert
        	
        for(Id accountId : accountId_lst) {
            if(account_cno.containsKey(accountId)) {
                account acc = new account();
                acc.Id = accountId;
                acc.Number_of_Contacts__c = account_cno.get(accountId);
                account_lst.add(acc);
            }
            
        }
        upsert account_lst;
    }

}



@isTest
public class AccountProcessorTest {
    
    @isTest
    public static void testFunc() {
        account acc = new account();
        acc.name = 'MATW INC';
        insert acc;
        
        contact con = new contact();
        con.lastname = 'Mann1';
        con.AccountId = acc.Id;
        insert con;
        contact con1 = new contact();
        con1.lastname = 'Mann2';
        con1.AccountId = acc.Id;
        insert con1;
        
        
        List<Id> acc_list = new List<Id>();
        acc_list.add(acc.Id);
        Test.startTest();
        	AccountProcessor.countContacts(acc_list);
        Test.stopTest();
        List<account> acc1 = new List<account>([select Number_of_Contacts__c from account where id = :acc.id]);
        system.assertEquals(2,acc1[0].Number_of_Contacts__c);
    }

}