How to record transaction: Employee paid a large vendor invoice with their personal credit card

one way is to enter the bill, then use pay bills and pay it - do not print the check

hand write the check payable to the employee

"I'm reconciling for last fiscal year, so in the case of checks written outside that period that were cashed in that period, I need to figure out how to move money between the accounts properly without changing anything that was declared on the taxes for the previous fiscal year (does that make any sense?"

Just because something cleared or didn't clear, does not apply. You don't Move anything. You issued a check, and whether or not it is cashed is part of Banking, not part of your perspective. You pay my in Nov and I don't cash it until Jan = you Incurred Subcontractor labor in Nov.

"The reimbursement was included on the employee's normal paycheck, but for some reason, the bill payment was recorded on one of the company credit cards.  Since I'm reconciling bank accounts, that transaction can't stay and I remain in balance, so how would I go about retroactively correcting this (I'm sure this isn't the only one of these I'll run into, as I've only gotten 6 months in.)"

You need a Credit Card Credit, and it gets offset to whatever is listed on the payment to the employee, because it appears your expense is entered twice using two different methods.

Or, delete the credit card  charges if it is a Duplicate entry for what was used for the payment to the employee.

Or, edit the employee payment (if this was a check and not a Paycheck), and it should not be Expense; it would be your CC account, to offset that entry error there and remove the Balance from the CC, leaving the details intact.

For QB desktop:

This worked for me:

Create a Reimbursement Account (Other Current Liability). Enter the vendor bill normally (use normal account, amount, customer: job, class). Go to Pay Bills, select the bill in question and click Discount Info. and enter the amount of the vendor bill as a discount and use the Reimbursement Account (the one you created in first step)  - this "pays" the bill" to the vendor. Issue a check to the employee ( or whoever paid the invoice) and use the Reimbursement Account as the account. This pulls the expense through. This process generates a journal entry in Accounts Payable for the Vendor (increase) which is offset by the payment to the employee for the equal amount paid to the vendor (decrease). The net effect is that the Reimbursement account has a zero balance. The expense is shown in P&L associated with the Vendor.

You need to look at WHY the employee paid the large company expense on personal credit card. It's not good business. I have seen this happen often. Different reasons ...to accrue points/cash back on credit cards which would then essentially be income or to get large sum of cash without a cash advance rate on their credit card or various other reasons. Either way, it should not happen and a determination as to why!

So I assume you entered the bill for the freight.  You need to set up the employee as a vendor.  You will have to use some different form of their name so that is not a duplicate list entry. 

Enter a vendor bill toggling the credit field at the top of the enter bill field and enter the amount that the employee paid.  Go to pay bills.  Select the vendor and apply the credit that shows at the bottom of that window.  If not paid in full, swipe through the amount paid on the vendor screen and type in the amount that the employee paid.

Enter a bill for the amount that the employee paid with the employee as the vendor.