As the others have commented: It's recommended to create a calendar table. At minimum, you will need to know your Holidays and/or closure dates. I would turn this into two separate functions but below I'll show you as a single query.

Also, consider evaluating if there are slight changes for January. Example - If you run this on 1-15-2023, will they want the start date to run into the prior year?

1. The key is to create a CTE to generate dates.
2. Then another CTE that references the first to filter out your Holidays.
3. Return the TOP 2 or TOP 3 while filtering out Weekdays. The built-in DATEPART() function, using DW as your datepart, will give you an integer for the week; 1 = Sunday 7 = Saturday.

Example:

DECLARE @startDate DATE = GETDATE();
DECLARE @firstBusinessDay DATE;
DECLARE @lastBusinessDay DATE;

--Add Holidays If needed, Highly recommended that this become an actual table that is managed
DECLARE @Holidays TABLE (
    HolidayDate Date
)
--Holiday sample dates
INSERT INTO @Holidays VALUES 
    ('2023-01-02'),
    ('2023-04-07'),
    ('2023-05-29'),
    ('2023-07-04'),
    ('2023-09-04'),
    ('2023-11-23'),
    ('2023-11-24'),
    ('2023-12-25'),
    ('2023-12-26')
;WITH CTE
AS
(
    SELECT 
        EOMONTH(@startDate,-1) AS [day], --Anchor for last day of prior month
        1 as [ct]
    UNION ALL --Union between Anchor member and Rescursive members
    SELECT 
        DATEADD(DAY,-1,[day]), --Add each prior date
        [ct]+1 --We use this to stop the recursion in the WHERE clause
    FROM CTE  --Reference the current CTE to create an recursion
    WHERE [ct] < 5 --We will never need more than 5 dates to work with, so we stop the recursion
), CTE_WEEKDAYS
AS
(
    SELECT TOP 2 [day] FROM CTE WHERE DATEPART(DW,[day]) NOT IN (1,7) --Only days that are weekdays
)
SELECT TOP 1 @firstBusinessDay = [day] FROM CTE_WEEKDAYS WHERE [day] NOT IN (SELECT HolidayDate FROM @Holidays) ORDER BY [day] ASC  --2nd to last calendar day of the prior month, ignoring Holidays
;WITH CTE
AS
(
    SELECT 
        EOMONTH(@startDate) AS [day], --Anchor for last day of prior month
        1 as [ct]
    UNION ALL --Union between Anchor membe and Rescursive members
    SELECT 
        DATEADD(DAY,-1,[day]), --Add each prior date
        [ct]+1 --We use this to stop the recursion in the WHERE clause
    FROM CTE WHERE [ct] < 5 --We will never need more than 5 dates to work with, so we stop the recursion
), CTE_WEEKDAYS
AS
(
    SELECT TOP 3 [day] FROM CTE WHERE DATEPART(DW,[day]) NOT IN (1,7) --Only days that are weekdays
)
SELECT TOP 1 @lastBusinessDay = [day] FROM CTE_WEEKDAYS WHERE [day] NOT IN (SELECT HolidayDate FROM @Holidays) ORDER BY [day] ASC --3nd to last calendar day of the current month, ignoring Holidays
SELECT @firstBusinessDay, @lastBusinessDay --Start and End days for the period