This is very similar to the problem of generating a Bitcoin key whose address has some predetermined content, called a 'vanity' address by analogy to the 'vanity' automobile license plates available in many (most?) jurisdictions at extra cost, except Bitcoin needs to guess approximately 6 bits for each desired character in the result address. See other stack for Qs on that.
You could presumably modify the Bitcoin vanitygen program to handle your case, since the hard part is the point multiplication; hashing is relatively cheap and base58 negligible.
It's not clear if you really want fixed bytes (and possibly your byte size, although since you didn't think to specify you probably only know about PC-class machines that use 8-bit bytes) or since the example you gave is all valid hexadecimal digits if you actually mean fixed digits in the hex representation often used to display and sometimes to store or transport EC keys (and other things in modern crypto).
If you want to fix 11 hex digits that's 44 bits; based on the rough estimates at https://en.bitcoin.it/wiki/Vanitygen it probably takes less than a day or at most a few days on reasonable hardware.
If you want to fix 11 bytes of 8 bits that's much costlier, for example a hundred thousand years if you use a million PC-class machines, somewhat less if you have custom hardware designed and manufactured for you.