With 36 hours to prepare, a level 20 wizard of the School of Necromancy can flip 1392 non-illusory birds in 3 rounds.
The answer provided by NautArch is pretty much unbeatable given the question as posed, but let's assume your mother-in-law has truesight and will therefore be thoroughly unimpressed by any illusory birds you flip at her. Let's see how many real (or at least non-illusory) hands we can make.
Here's the plan:
Let's start the day before the big event. First, you, a level 20 wizard of the School of Necromancy1, cast simulacrum on yourself, making a copy that is missing only a single 7th-level spell slot. Then you use your other 7th level slot to cast magnificent mansion, inside which you will serve dinner tomorrow. The mansion supplies you with 100 servants whose appearance you decide. Let's assume your imagination is limited to only creatures you know to exist, so you decide to make each servant look like a marelith with 6 hands (because if there's no limit then this answer ceases to be interesting).
With all your remaining 3rd and higher level spell slots, you head over to the local cemetery and cast animate dead. The locals may initially object, but I'm sure that once you explain how awful your mother-in-law is, they'll understand your need. Make sure to emphasize that you're only borrowing their ancestors' sacred remains, and they can have the corpses back tomorrow. Remember to check each corpse for 2 intact middle fingers before animating it! Once this is complete, you cast Leomund's tiny hut as a ritual in the cemetery and camp for the night with your simulacrum, taking a long rest.
In the morning, you awake refreshed and take stock of the 78 animated undead you created yesterday. Then, you continue the work you began last evening, using all your 3rd through 6th level spell slots and your 8th level spell slot, as well as all your simulacrum's 3rd through 8th level spell slots, to animate an additional 138 undead. (Did I mention you should do this in large city with an ample supply of corpses?) With all 216 animated corpses in tow, you return to your mansion to cast a 7th level conjure animals into a 7th-level glyph of warding. Your advance preparations are now complete. (Presumably the 100 servants in the mansion can handle the preparation of the meal itself, leaving you to focus on the important matters.)
When your mother-in-law arrives, you and your simulacrum each cast mage hand, using the hands to present your beautifully plated dish (and also ensuring you don't need to waste an action later casting mage hand once the action starts). When she takes a bite and makes her rude comment, your glyph triggers, summoning 24 flying monkeys.2 At the same time, you and your simulacrum each cast conjure animals at 9th level (using both your concentration), bringing the total number of flying monkeys to 88. Then, you and your simulacrum each use your next actions to move your mage hands into position, and your next actions after that to cast chill touch, and finally use both of each of your own hands to flip the bird at her (free actions on your turns), completing the springing of the trap over the course of 3 rounds (or 2 rounds for a Wizard 17/Sorcerer 3 with Quickened Spell).
So, let's get the final count of birds flipped:
- 100 near-transparent marelith-resembling servants from the mansion, 6 hands each: 600
- 216 animated undead humanoids, each with 2 hands: 432
- 88 conjured flying monkeys, each with 2 hands and 2 feet capable of flipping the bird: 352
- you and your simulacrum, each with 2 hands and 2 bird-flipping cantrips: 8
Add these all up, and the total number of non-illusory hands (and feet) awaiting your bird-flipping command is 1392. For the sake of completeness, note that you will also be dealing up to 8d8 necrotic damage to your mother-in-law with the two chill touch spells, which adds some injury to this insult.
Since you still have your 1st and 2nd level spell slots left over, you and your simulacrum may as well spend the 4th round casting suggestion on your mother in law, suggesting that she cave to peer pressure and flip herself the bird. If she fails the saving throw, you can add 2 hands to the total, but this isn't guaranteed. Similarly, you might be able to use Charm Person to convince some of the locals to join the party.
With all of these monkeys and undead in your mansion, it's a good thing it will cease to exist in a few hours, because you do not want to clean up the aftermath of this event. Remember that you control the floor plan of the mansion. Make sure to design a dining room that can comfortably accommodate 369 creatures (although a high ceiling should be sufficient to accommodate the flying monkeys).
Oh, and don't forget to tell all the undead afterward to report back to the town for re-burial. Sure, they're still animated, but I'm sure the villagers can sort it all out.
1Credit to user ttolluz for adding in the School of Necromancy, whose Undead Thralls feature adds one additional undead humanoid per casting of Animate Dead.
2I don't have the exact stats for a flying monkey available, but I'm willing to make some assumptions about their ability to flip the bird.