240v plug, converter or transformer for EU kitchen appliance in the US

The problem with a transformer will be the amperage, to make it simple power on the primary is equal to power on the secondary or if 1500w @ 240v or 6.25 amps.

Cutting the voltage in half doubles the amperage to do the same job 120v load would be 1500 and that would be 12.5 amps. But an ac motor increasing frequency will draw ~20% more or 15 amps so it would require a 20 amp circuit and if other things are on the circuit it may trip the breaker on start up.

Using the existing oven receptacle the breaker would normally need to be reduced in size and the receptacle changed to be code compliant.

This might work if the electronics convert to DC some work that way. Where troubles arise, ac motors and a 10 Hz frequency shift may not work well unless the motor was designed for 50 / 60 Hz, the problem comes with a higher frequency. AC motors spin faster and draws more current causing the motor to overheat in some cases (I believe it is 20% more). The excessive current draw can damage or shorten the life of any electronics that control speed (SCR’s and Triacs) due to the higher current.

Going the other way from 60 to 50 Hz usually is safer but the motor spins slower and may not make speed for a centrifugal clutch to shift into the high winding.

So you can see it may be possible however it also may damage your pride and joy kitchen equipment. There are plug in transformers that can do the voltage conversion around $75.00 and up.

There are other devices called inverters or variable frequency drives than can step up the voltage and change the frequency. the small ones that would do the job start in the low $300 range but then there is the box to put it in and receptacles and programming. You can see this method is expensive but normally out of reach for most consumers but would work if you want to do that much work.

As far as frequency difference and motor speed, note what Ed Beal says.

Your worst case play is to get an "online" UPS from the European world. Most UPS's just pass-through AC power as they get it, while also using it to charge the battery. Only when power fails do they switch to battery+DC-AC inverter. For an online UPS, it's different. They convert all incoming power to DC, and the DC-AC inverter is used at all times - so a switch to battery is seamless and instant. As a result, it always outputs 50Hz even if the input is 60Hz. But that only matters if your motor cares. Let's see the nameplate.

Use the existing 240v outlet in my kitchen with an adapter. That would require calling an electrician to do the work.

No. That "existing 240V outlet" is for the oven/range. It has a 40A or 50A circuit breaker. That appliance is designed to fail in a way that would trip a 13A or 16A breaker. If a 40A breaker were "protecting" it, the breaker wouldn't even notice while the appliance burns.

We really need to see the nameplate on the appliance, but I'm guessing it's between 6 and 13 amps at 230V. It would take a 15A (or tops, 20A) breaker to provide adequate protection.

If you don't plan to use an electric range or oven, it is possible to change both the socket and the breaker to 15-20A, using the existing wiring in the walls. While you're at it, you might want to use something like Legrand Wiremold surface conduit to bring the socket somewhere more practical. Code requires kitchen appliances have cords no longer than 2 feet, and strongly discourages extension cords.

Keep in mind the 240V outlet will NOT be GFCI, which means if it falls into the sink, you are dead. So don't do insane things like draping an extension cord across the sink area, or having socket+cord such that it could get near the sink. However if you are one of the several US states that has adopted NEC 2020, your 240V circuit will need to be GFCI - upside no worries; downside requires $80 GFCI breaker.

Use a transformer. Based on the fact that the KM096 can draw up to 1500W, I'm thinking 3000W to be safe

No. The problem is, EU wall sockets can deliver 3000W, But USA wall sockets cannot.

The downside of our safer 120V is that less power is available, and for many EU heating appliances, that's just not enough. That's one thing that endears EU appliances to Americans, actually; they work better because they have more power to work with.

What you'll need to do is have an electrician add a dedicated 240V kitchen countertop circuit. You won't find too many electricians who will be will be willing to do this without that $80 GFCI breaker, which will require 2 full spaces in the panel. If your panel is full, that will require some rearranging which will be extra work and $$$, which may also expose other flaws which also need addressing. Upshot, this could snowball into $500, or even as much as $2000 of work if the electrician arrives and finds a hopelessly miswired FPE main panel, for instance.

You may get very lucky and find an existing "Multi-Wire Branch Circuit" configured in a way that lets the electrician swap in a dual-voltage receptacle and you're good to go. That would be an easy $100-ish job.