You can use np.where and date.between to allocate the pre and post status and group by the same and websites and find mean.
In one line(though not so readable):
df['date']=pd.to_datetime(df['date'])
df.groupby([np.where(df['date'].between('1/1/2021','1/3/2021'),'pre'
,'post'),'website'])['amount_views'].mean().to_frame('mean')
Step by step (more readable):
df['date']=pd.to_datetime(df['date'])
df['status']=np.where(df['date'].between('1/1/2021','1/3/2021'),'pre','post')
df.groupby(['status','website'])['amount_views'].mean().to_frame('mean')
mean
status website
post a 13.500000
b 9.500000
pre a 16.666667
b 8.000000
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